"No, I would not. And here's why. There is this creeping attempt, there is this attempt to gradually ease Sharia law and the Muslim faith into our government. It does not belong in our government. This is what happened in Europe. And little by little, to try and be politically correct, they made this little change, they made this little change. And now they've got a social problem that they don't know what to do with hardly.
The question that was asked that 'raised some questions' and, as my grandfather said, 'I does not care, I feel the way I feel.'"
Setting aside how asinine and ridiculous his "feelings" are, he's simply not allowed by the US Constitution to act on them. This is not in dispute The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious test shall ever be required as a qualification to any office or public trust under the United States.
You just can't refuse to appoint people on the basis of their religion without changing the constitution, which explicitly precludes doing that.