It's not about that, though
That was, in a loose sense, my inspiration for wondering about this. I think the question of a President perceived as a Muslim is interesting in its own right.
My initial thinking is that it would, at a minimum, induce cognitive dissonance, which might be an opening to get many Muslims to reconsider their narrow view of the United States as a "Christian Crusader". It might put some substance behind our claim to religious tolerance. In an authoritative social system, would such an American President gain authority?