Proof from Holy Scripture and the Catechism of the Catholic Church
The Catechism of the Catholic Church states:
I. THE JUDGMENT OF CONSCIENCE
1777 Moral conscience, present at the heart of the person, enjoins him at the appropriate moment to do good and to avoid evil. It also judges particular choices, approving those that are good and denouncing those that are evil. It bears witness to the authority of truth in reference to the supreme Good to which the human person is drawn, and it welcomes the commandments. When he listens to his conscience, the prudent man can hear God speaking.
1782 Man has the right to act in conscience and in freedom so as personally to make moral decisions. “He must not be forced to act contrary to his conscience. Nor must he be prevented from acting according to his conscience, especially in religious matters.”
1783 Conscience must be informed and moral judgment enlightened. A well-formed conscience is upright and truthful. It formulates its judgments according to reason, in conformity with the true good willed by the wisdom of the Creator. The education of conscience is indispensable for human beings who are subjected to negative influences and tempted by sin to prefer their own judgment and to reject authoritative teachings.
1784 The education of the conscience is a lifelong task. From the earliest years, it awakens the child to the knowledge and practice of the interior law recognized by conscience. Prudent education teaches virtue; it prevents or cures fear, selfishness and pride, resentment arising from guilt, and feelings of complacency, born of human weakness and faults. The education of the conscience guarantees freedom and engenders peace of heart.
1785 In the formation of conscience the Word of God is the light for our path, we must assimilate it in faith and prayer and put it into practice. We must also examine our conscience before the Lord’s Cross. We are assisted by the gifts of the Holy Spirit, aided by the witness or advice of others and guided by the authoritative teaching of the Church.
1786 Faced with a moral choice, conscience can make either a right judgment in accordance with reason and the divine law or, on the contrary, an erroneous judgment that departs from them.
1787 Man is sometimes confronted by situations that make moral judgments less assured and decision difficult. But he must always seriously seek what is right and good and discern the will of God expressed in divine law.
1790 A human being must always obey the certain judgment of his conscience. If he were deliberately to act against it, he would condemn himself. Yet it can happen that moral conscience remains in ignorance and makes erroneous judgments about acts to be performed or already committed.
1791 This ignorance can often be imputed to personal responsibility. This is the case when a man “takes little trouble to find out what is true and good, or when conscience is by degrees almost blinded through the habit of committing sin.” In such cases, the person is culpable for the evil he commits.
1792 Ignorance of Christ and his Gospel, bad example given by others, enslavement to one’s passions, assertion of a mistaken notion of autonomy of conscience, rejection of the Church’s authority and her teaching, lack of conversion and of charity: these can be at the source of errors of judgment in moral conduct.
1794 A good and pure conscience is enlightened by true faith, for charity proceeds at the same time “from a pure heart and a good conscience and sincere faith.” . . .
In short, Catholics hold that a properly formed conscience is necessarily brought about by the input of the Bible and the Church. It can’t be separated from that. It’s not purely an interior phenomenon. Protestantism, by rejecting the infallibility of the Church, in effect detached the conscience from being (in the final analysis) in accord and in harmony with Church teaching. That was the wrong turn. The notion of “conscience formed by the guidance of the Church” is entirely a scriptural one, as I shall now prove.
Our consciences are clearly formed by many outside influences: upbringing, culture, friends, moral teachings inculcated in the normal course of life, and the Bible itself and whatever ecclesiastical authority we attempt to abide by. The Catholic argues that the Church is fundamental in forming an individual conscience, and that the two should not and will not conflict. If it falls back completely on the individual, on the other hand (as Protestant premises require), we have the same unsolvable problems we always run into: what to do when opinions contradict; both claim to be biblical, etc.?
The Apostle Paul doesn’t relegate doctrine to conscience alone (theoretically tied to the Bible alone). To the contrary, Paul is quite clear that there is one doctrine, one truth, one “faith” (“the faith”) and that this is in line with the Church (not — ultimately — logically or epistemologically separate from the Church, as in Protestantism). Obviously, whatever the individual arrives at through conscience, St. Paul would say has to be in line with “the faith,” “the truth,” “tradition,” “the gospel,” etc. And thus we are right back to unavoidable, binding Church authority:
Acts 16:4 (RSV) As they went on their way through the cities, they delivered to them for observance the decisions which had been reached by the apostles and elders who were at Jerusalem.
Romans 2:8 but for those who are factious and do not obey the truth, but obey wickedness, there will be wrath and fury.
Romans 16:17 . . . take note of those who create dissensions and difficulties, in opposition to the doctrine which you have been taught; avoid them.
Galatians 5:7 You were running well; who hindered you from obeying the truth?
Ephesians 3:10 . . . through the church the manifold wisdom of God might now be made known to the principalities and powers in the heavenly places.
Colossians 1:23 provided that you continue in the faith, stable and steadfast, . . .
Colossians 2:7 rooted and built up in him and established in the faith, just as you were taught, abounding in thanksgiving.
1 Thessalonians 2:13 And we also thank God constantly for this, that when you received the word of God which you heard from us, you accepted it not as the word of men but as what it really is, the word of God, which is at work in you believers.
2 Thessalonians 3:6 . . . the tradition that you received from us.
1 Timothy 1:2 To Timothy, my true child in the faith: . . .
1 Timothy 2:4 who desires all men to be saved and to come to the knowledge of the truth.
1 Timothy 3:15 if I am delayed, you may know how one ought to behave in the household of God, which is the church of the living God, the pillar and bulwark of the truth.
1 Timothy 4:3 . . . those who believe and know the truth. (cf. 4:1; 4:6; 5:8; 6:10, 12, 21)
2 Timothy 1:13-14 Follow the pattern of the sound words which you have heard from me . . . guard the truth which has been entrusted to you by the Holy Spirit who dwells within us. (cf. 2:18, 25; 3:7-8; 4:4)
2 Timothy 2:2 And what you have heard from me before many witnesses entrust to faithful men who will be able to teach others also.
Titus 3:15 . . . Greet those who love us in the faith. Grace be with you all. (cf. 1:1)
1 Peter 3:20-21 who formerly did not obey, when God’s patience waited in the days of Noah, during the building of the ark, in which a few, that is, eight persons, were saved through water.  Baptism, which corresponds to this, now saves you, not as a removal of dirt from the body but as an appeal to God for a clear conscience, through the resurrection of Jesus Christ,
The same Peter presupposes “the truth”: an objective thing that can be known and accepted:
1 Peter 1:22 Having purified your souls by your obedience to the truth for a sincere love of the brethren, love one another earnestly from the heart.
2 Peter 1:12 Therefore I intend always to remind you of these things, though you know them and are established in the truth that you have.
2 Peter 2:1-2 But false prophets also arose among the people, just as there will be false teachers among you, who will secretly bring in destructive heresies, even denying the Master who bought them, bringing upon themselves swift destruction. And many will follow their licentiousness, and because of them the way of truth will be reviled.
And he makes other casual dogmatic statements, suggesting a strong Church authority and a binding tradition in addition to conscience and the Bible:
2 Peter 2:21 . . . the holy commandment delivered to them.
2 Peter 3:1-2 This is now the second letter that I have written to you, beloved, and in both of them I have aroused your sincere mind by way of reminder; that you should remember the predictions of the holy prophets and the commandment of the Lord and Savior through your apostles.
The author of Hebrews, like Peter and Paul, maintains the “three-legged stool” view of the Catholic Church through the ages, not the private judgment / absolute supremacy of individual conscience / sola Scriptura position of Protestantism:
Hebrews 10:26 For if we sin deliberately after receiving the knowledge of the truth, there no longer remains a sacrifice for sins,
Hebrews 13:7 Remember your leaders, those who spoke to you the word of God; consider the outcome of their life, and imitate their faith.
St. Paul and other writers, again, do not disconnect an informed Christian conscience from the guidance of the Church:
Acts 23:1, 6-8 And Paul, looking intently at the council, said, “Brethren, I have lived before God in all good conscience up to this day.” . . .  But when Paul perceived that one part were Sad’ducees and the other Pharisees, he cried out in the council, “Brethren, I am a Pharisee, a son of Pharisees; with respect to the hope and the resurrection of the dead I am on trial.”  And when he had said this, a dissension arose between the Pharisees and the Sad’ducees; and the assembly was divided.  For the Sad’ducees say that there is no resurrection, nor angel, nor spirit; but the Pharisees acknowledge them all.
Paul here shows exactly what he means by having a “good conscience.” It was formed in the context of tradition. He appeals to a doctrinal school (Pharisaism) and identifies himself with it (over against the Sad’ducees, who denied the resurrection, and angels). That was what it meant to live “before God in all good conscience.” It can’t be divorced from binding doctrines. Yet Protestantism stresses the radically autonomous individual conscience and the corresponding notion that there are no binding decrees of the Church that can be considered infallible and unquestionable. It’s exactly the same (over against Protestantism) in the next chapter of Acts:
Acts 24:14-16 But this I admit to you, that according to the Way, which they call a sect, I worship the God of our fathers, believing everything laid down by the law or written in the prophets,  having a hope in God which these themselves accept, that there will be a resurrection of both the just and the unjust.  So I always take pains to have a clear conscience toward God and toward men.
Having a “clear conscience toward God” is directly tied, in Paul’s mind, to observance of the Law (a binding thing), and doctrines like the general resurrection. Not just the Church, but even secular rulers are to be obeyed “for the sake of conscience”:
Romans 13:1-7 Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God.  Therefore he who resists the authorities resists what God has appointed, and those who resist will incur judgment.  For rulers are not a terror to good conduct, but to bad. Would you have no fear of him who is in authority? Then do what is good, and you will receive his approval,  for he is God’s servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain; he is the servant of God to execute his wrath on the wrongdoer.  Therefore one must be subject, not only to avoid God’s wrath but also for the sake of conscience.  For the same reason you also pay taxes, for the authorities are ministers of God, attending to this very thing.  Pay all of them their dues, taxes to whom taxes are due, revenue to whom revenue is due, respect to whom respect is due, honor to whom honor is due.
St. Paul ties in conscience with “the truth” and “doctrine” and “divine training” and “the faith”:
2 Corinthians 4:2 We have renounced disgraceful, underhanded ways; we refuse to practice cunning or to tamper with God’s word, but by the open statement of the truth we would commend ourselves to every man’s conscience in the sight of God.
1 Timothy 1:3-7 As I urged you when I was going to Macedonia, remain at Ephesus that you may charge certain persons not to teach any different doctrine,  nor to occupy themselves with myths and endless genealogies which promote speculations rather than the divine training that is in faith;  whereas the aim of our charge is love that issues from a pure heart and a good conscience and sincere faith.  Certain persons by swerving from these have wandered away into vain discussion,  desiring to be teachers of the law, without understanding either what they are saying or the things about which they make assertions.
1 Timothy 3:9 they must hold the mystery of the faith with a clear conscience.
1 Timothy 4:1-2 Now the Spirit expressly says that in later times some will depart from the faith by giving heed to deceitful spirits and doctrines of demons,  through the pretensions of liars whose consciences are seared,
Titus 1:13-15 This testimony is true. Therefore rebuke them sharply, that they may be sound in the faith,  instead of giving heed to Jewish myths or to commands of men who reject the truth.  To the pure all things are pure, but to the corrupt and unbelieving nothing is pure; their very minds and consciences are corrupted.
The writer of Hebrews also follows suit:
Hebrews 10:22-23 let us draw near with a true heart in full assurance of faith, with our hearts sprinkled clean from an evil conscience and our bodies washed with pure water.  Let us hold fast the confession of our hope without wavering, for he who promised is faithful;
Hebrews 13:17-18 Obey your leaders and submit to them; for they are keeping watch over your souls, as men who will have to give account. Let them do this joyfully, and not sadly, for that would be of no advantage to you.  Pray for us, for we are sure that we have a clear conscience, desiring to act honorably in all things.
We see again and again, therefore, that the Catholic approach, as stated in the Catechism, is also the biblical one, and that this biblical / Catholic perspective is contrary to the ultimately autonomous conscience of Protestantism.