Well, the Bill of Rights, rather.
1st Amendment, freedom of religion, speech, political opinion, etc.
You could argue that he wouldn't be able to "deny to any person within its jurisdiction the equal protection of the laws" (14th Amendment) but in actual practice, it is up to the jurisdiction of the supreme court.
However, laws were already passed (which had to pass through Congress and Senate) that would allow the president to bar entry for certain aliens:
(f) Suspension of entry or imposition of restrictions by President
Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate.
And as I stated earlier, the teachings of Islam could be quoted as being detrimental to the interests of the United States. Ultimately, it's up to the executive branch how to enforce the immigration laws, but I doubt what Annoying Orange said is going to disqualify him.
https://www.law.cornell.edu/uscode/text/8/1182 (f)
