. . . With My Biblical and Historical Refutations
I have always been fascinated and revolted by this sort of thinking, which is simultaneously so unbiblical and anti-historical in nature, whereas ironically the same opponents of Christmas have viewed themselves as exceptionally “biblical” and “Christian” in a way that we lowly, pagan Christmas celebrants allegedly are not. Much of this thinking was at root, anti-Catholic as well, or even anti-Anglican or anti-Lutheran (anti- any Christians group other than extreme Calvinism / Puritanism).
There was also a strain of opposition that made perfect sense from a Christian and biblical perspective: in cases where the celebration of Christmas had become truly raucous, drunken, and even sexualized, and had little or nothing to do with the true meaning of Christmas (sort of like what New Year’s Eve has become). I’m not interested in documenting that because I agree with it. Nor do I have any interest in purely personal / emotional Scrooge-like opposition.
I deal here with purely theological opposition. It’s good and worthwhile to address such a topic as an apologist, even though this fringe view is held today by a miniscule number of Christians. Sources are listed at the end by number, so they can be referred to for each excerpt. All citations will be indented. I have removed the over-abundance of bolding.
I. Antipathy to Holy Days
The regulative principle of worship has clear implications for those who want to promote the celebration of Christmas. The Regulative Principle forces those who celebrate Christmas to prove from Scripture that God has authorized the celebrating of such a day. This, in fact, is impossible. 
The reason that Christmas became a church holy day has nothing to do with the Bible. 
[C]oncerning festival days findeth that in the explication of the first head of the first book of discipline it was thought good that the feasts of Christmas, Circumcision, Epiphany, with the feasts of the Apostles, Martyrs, and Virgin Mary be utterly abolished because they are neither commanded nor warranted by Scripture and that such as observe them be punished by Civil Magistrates. Here utter abolition is craved and not reformation of abuses only and that because the observation of such feasts have no warrant from the word of God. 
I would to God that every holy day whatsoever besides the Lord’s day were abolished. 
The Bible teaches both about holy days and the veneration of saints, including certainly a much-higher worship and adoration of our Lord Jesus (which I need not even defend):
Bible on the Veneration of Angels & Men [9-10-15]
II. Supposed Late Historical Arrival and Purely Pagan Origin (Saturnalia and Sol Invictus)
[T]he day was first set apart for this purpose by the authority of the bishop at Rome, toward the close of the fourth century, or early in the fifth. 
It is also noted that Scripture never commands the celebration of this day and that there is no evidence that Christ and the Apostles ever celebrated this day — in fact, this sycretism of paganism and “Babylonian” Christianity was not first celebrated until 354 A.D. (when December 25 was chosen, in accord with the Pagan feast of Saturnalia, as the day of “celebration”). 
Not true: we have records of Christian celebrations of Christmas on December 25th, from Pope St. Telesphorus (c. 125-136), the seventh bishop of Rome, St. Theophilus (AD 115-181), bishop of Caesarea, and St. Hippolytus (170-240). The Roman feast of Saturnalia was celebrated from December 17-23, not the 25th. Sol Invictus was instituted by the Roman emperor Aurelian on Christmas Day, 274 AD, 138-149 years after Christians are documented to have celebrated Christmas on December 25th. The first historical evidence of the Christian Church “Christianizing” pagan festivals is in a letter of Pope St. Gregory the Great from 601 AD. The first suggestion that the date of Christmas was deliberately determined to be the same as pagan feasts occurred in the 12th century AD in a marginal note in the writings of the Syriac bishop Dionysius bar-Salibi.
III. Supposedly Ignorant and Non-Biblical Determination of the Date of Jesus’ Birth
The truth is, the day of Christ’s nativity has been irrecoverably lost. Had this date been designed for special religious veneration, its date would have been preserved in the Holy Record, and a divine command given for its proper observance. The absence both of the date and command, makes it as clear to us as a sunbeam, that the natal day of our Saviour, even were it known, should not be honored by any religious observance whatsoever. 
The Bible does not give the date of Christ’s birth. 
Moreover, who knows when, in what month, He was born? The Bible is silent thereon. 
This is manifestly and demonstrably false. The Bible states that Jesus was six months younger than John the Baptist (Lk 1:36). So all we need to know is the date of the conception of John, which was revealed to his father Zechariah by the archangel Gabriel (Lk 1:5-24). Zechariah was a priest from the class of Abijah (Lk 1:5). These classes served for a week in the temple, two times a year (see 1 Chr 24:10; cf. 24:1-19). German Catholic theologian Josef Heinrich Friedlieb (1810-1900) documented that Zechariah’s class would have been serving in the temple in the second week of the Jewish month Tishri (late September). That was simply an inquiry directly stimulated or inspired by the biblical information.
While thus serving, the Archangel Gabriel revealed to Zechariah that his wife Elizabeth was to bear a son (Lk 1:5-24). That means that John the Baptist was conceived in late September, with his birth arriving at the end of June, 3 BC. Thus, the Church celebrates the Nativity of St. John the Baptist on June 24th. These dates aren’t just pulled out of hats with rabbits, too. Since Scripture tells us that Jesus was six months younger, then he was conceived in late March and born c. 25 December. Therefore, we do know at least the month and approximate date from Holy Scripture.
We don’t need an express command to “worship Jesus on his birthday.” The Bible never states that it must give explicit indications of everything we are permitted to believe. God expects us to use our brains and reason. But since holy days are quite biblical (derived from the Jewish feasts and also the Sabbath and Sunday as the Lord’s day), all we need is permission to worship Jesus, which is certainly in the Bible:
Matthew 14:33 (RSV) And those in the boat worshiped him, saying, “Truly you are the Son of God.” (cf. 2:2; 2:11; 8:2; 9:18; 15:25; 20:20)
Matthew 28:9 And behold, Jesus met them and said, “Hail!” And they came up and took hold of his feet and worshiped him. (cf. 28:17; Mk 5:6; Lk 24:52; Jn 5:23; 9:38)
Philippians 2:9-11 Therefore God has highly exalted him and bestowed on him the name which is above every name,  that at the name of Jesus every knee should bow, in heaven and on earth and under the earth,  and every tongue confess that Jesus Christ is Lord, to the glory of God the Father.
Hebrews 1:6 And again, when he brings the first-born into the world, he says, “Let all God’s angels worship him.” (cf. 7:26; Acts 2:33; 5:31; Rev 5:11-14; 7:9-12)
IV. Is it Wrong to Celebrate and “Make Merry” About the Birth of Jesus and God Becoming Man? No! Clear Biblical Models
[T]he natal day of our Saviour, even were it known, should not be honored by any religious observance whatsoever. 
Nowhere in the Bible are we commanded to celebrate Christmas. 
But, says someone, Christmas is the time when we commemorate the Savior’s birth. It is? And WHO authorized such commemoration? Certainly God did not. The Redeemer bade His disciples “remember” Him in His death, but there is not a word in scripture, from Genesis to Revelation, which tells us to celebrate His birth. . . . It is without reason that the only “birthday” commemorations mentioned in God’s Word are Pharaoh’s (Gen. 40:20) and Herod’s (Matt. 14:6)? . . . State plainly that you have been brought to see that “Christmas merrymaking” is entirely a thing OF THE WORLD, devoid of any Scriptural warrant; that it is a Romish institution, and now that you see this, you dare no longer have any fellowship with it (Eph. 5:11) 
How are we taught to act with regard to Jesus’ birth and Christmas, according to Scriptural models? We do definitely have those. An angel of the Lord told the shepherds in the field near Bethlehem: “I bring you good news of a great joy which will come to all the people; for to you is born this day in the city of David a Savior, who is Christ the Lord” (Lk 2:10-11). Then a “multitude” of angels were “praising God and saying, ‘Glory to God in the highest'” because of the birth of Jesus (Lk 2:13-14). The shepherds, after finding Jesus and worshiping Him, were “glorifying and praising God for all they had heard and seen, as it had been told them” (Lk 2:20). Likewise, the wise men, along with their worship, “rejoiced exceedingly with great joy” (Mt 2:10).
Se we see all of that: showing how both angels and men reacted to the birth of Jesus, but now the oh-so-pious Calvinists (well, an extreme fringe of them) want to come and tell us that it’s wrong and unbiblical and immoral — against God’s will — to celebrate the birth of Jesus our Lord and Savior and Redeemer? It’s not only ridiculous in the highest degree, but wicked and blasphemous to believe such a thing. These are natural human emotions, and we can and should express them on holidays, with celebrations and happiness and joy and merriment. This is how King David acted before the ark of the covenant, which contained God’s special presence:
2 Samuel 6:5, 14-16, 21 And David and all the house of Israel were making merry before the LORD with all their might, with songs and lyres and harps and tambourines and castanets and cymbals. . . .  And David danced before the LORD with all his might; . . .  So David and all the house of Israel brought up the ark of the LORD with shouting, and with the sound of the horn.  . . . King David leaping and dancing before the LORD . . .  And David said to Michal, “. . . I will make merry before the LORD.”
How much more do we and should we show joy and “make merry” and dance and play music to celebrate God becoming man? If we can’t be joyful and demonstrative about that (the Good News!), then we should about nothing at all.
V. Christmas Trees as Idols
“Thus saith the LORD, Learn not the way of the heathen, and be not dismayed at the signs of heaven; for the heathen are dismayed at them. For the customs of the people are vain: for one cutteth a tree out of the forest, the work of the hands of the workman, with the axe. They deck it with silver and with gold; they fasten it with nails and with hammers, that it move not. ” (Jer. 10:2-4). 
VI. Antipathy to Celebration of the Sacrifice of the Mass on Christmas, as Indicated by its Name / Supposed Idolatry
The word Christ-mass is enough to cause such as are studious of reformation to dislike what shall be known by a name so superstitious. Why should Protestants own any thing which has the name of Mass in it? How unsuitable is it to join Christ and Mass together? i.e., Christ and Antichrist. But what Communion has light with Darkness, and what concord hath Christ with Belial? 
By communicating with idolaters in their rites and ceremonies, we ourselves become guilty of idolatry; even as Ahaz, 2 Kings 16:10, was an idolater, eo ipso, that he took the pattern of an altar from idolaters. 
Why Transubstantiation Isn’t Idolatry At All [11-22-96]
Sacrifice of the Mass & Hebrews 8 (vs. James White) [3-31-04]
Catholic Mass: Can’t Possibly be Idolatrous [12-15-08]
St. Paul Was a Priest [9-15-15]
Luther Espoused Eucharistic Adoration [9-17-15]
Catholic Mass: “Re-Sacrifice” of Jesus? [11-19-15]
VII. No Worship or Christian Celebration That Isn’t Explicitly Sanctioned and Mentioned in Scripture is Permitted?
The scriptures, both by precept and example, forbid the use of any form of worship which is not ordained by God. Since Christmas has no biblical warrant, it should be rejected, even if there were no other reason to question it. 
Well, it does in fact have quite a bit of biblical warrant and indication, as shown in sections I, II, and IV above. But why does anyone think that the Bible teaches that everything must be explicit in the Bible to be believed by Christians in the first place? This is itself an unbiblical doctrine. I wrote in my article, “Explicit” Bible Proofs and Protestant Double Standards (2-12-16):
[T]he New Testament never mentions an “altar call”. It never has the typical “sinner’s prayer” of evangelicals. It doesn’t mention church buildings. It never uses the word “Trinity.” It never uses the frequently mentioned evangelical terminology of “personal relationship with Jesus.” It never lists its own books (the biblical canon comes from the authority and proclamations of the Catholic Church and tradition). It never teaches sola Scriptura, or the concept that the Bible is the only infallible source of authority. Yet –oddly enough and passing strange — this is one of the very “pillars” of the Protestant worldview.
Other beliefs or practices not explicitly mentioned in the Bible are Bible studies, separating young people during church services, and grape juice as an element to be consecrated for communion (rather than wine), “asking Jesus into one’s heart,” a “body of believers,” Scripture interpreting Scripture (the more clear helping to understand the less clear), agreeing on “essential” or “primary” doctrines and permitted relativism regarding “non-essential” or “secondary” doctrines, denominations (vs. the biblical “one Church”). Of course, this very idea that one must find explicit biblical proof for every doctrine or it can’t / mustn’t be believed (even with high selectivity or rank inconsistency) is not found in the Bible anywhere, either. It’s (irony of all ironies!) a mere tradition of men.
Some popular Protestant (and also often Catholic) words or phrases that do not appear in the Bible are rapture, invisible church, incarnation, virgin birth, holy communion, Lord’s prayer, Bible, original sin, fall of man, theology, go[ing] to church, grace alone, [total] depravity, unconditional election, limited atonement, irresistible grace, perseverance of the saints, spirituality, Scripture alone, pray for guidance, pray for direction, spiritual warfare, and sin nature. Faith alone only appears once:
James 2:24 You see that a man is justified by works and not by faith alone.
Protestants manage to believe all these things (or use these words) with no problem whatever. Why? Or, more specifically, why do they believe these things, which are absent from or non-explicit in the Bible, while giving Catholics misery for similar things, or else doctrines and practices with far more indication of various sorts than the things above, that Protestants accept? Why the double standard?
VIII. The Evil of Manger Scenes, Which Foster Idolatry?
Moreover, during the Christmas season numerous manger scenes and religious images are erected in public places, church buildings and homes. This multiplication of graven images is a blatant violation of the second commandment, which explicitly forbids making or using any pictorial representations of God. The second commandment prohibits the making of any images of God, including “pictures of Christ” in the manger. 
This is sheer nonsense. In fact, images of God Himself appear many times in Scripture, even in conjunction with worship, and crucifixes are in effect sanctioned:
“Graven Images”: Unbiblical Iconoclasm (vs. John Calvin) [Oct. 2012]
Is Worship of God Through an Image Biblical? (vs. Luke Wayne) [11-10-20]
2) The Regulative Principle of Worship and Christmas (book by Brian Schwertley, 1996).
3) “Christmas,” from The Reformed Presbyterian magazine, January, 1851.
4) “Christmas,” from The Associate Presbyterian Magazine, February, 1879.
5) The Acts of the General Assemblies of the Church of Scotland, December 10, Session 17, 1638, pp. 37-38
6) Pastor Greg L. Price, sermon: “Christmas Condemned by Christ.”
7) Martin Bucer, cited in William Ames, A Fresh Suit Against Human Ceremonies in God’s Worship (1633) , pp. 359-60.
8) Increase Mather’s Testimony Against that Prophane and Superstitious Custom of Christ-mass Keeping.
9) A Dispute Against English Popish Ceremonies, in Gillespie’s Works volume one, p. 80.
10) “Xmas (Christmas)”, by A. W. Pink.
11) Christmas: An Historical Survey Regarding Its Origins and Opposition to It, by Kevin Reed, 1995.