My paternal grandparents were immigrants to North Dakota. My grandfather from Denmark as an infant, though there’s a possibility that he was actually born in the United States. My grandmother came from Norway as a teenage girl, alone.
Some of the Muslim immigrants to North Dakota described in this article came at about the same time as my grandparents, or even perhaps a bit earlier.
I cannot think of a good reason for regarding my immigrant grandparents as “real Americans” while denying such a description — on purely religious grounds — to other immigrants who settled in the same place, farmed the same land, and served in the same army.