Planet of Hats
Donor
It's inherent in the role.Even if that's quite an obvious assumption (I have to admit that), is there something written specifically about it?
The Caliph is the Commander of the Faithful. If he's not one of the faithful, he can't very well be the Caliph, and the Muslim world would certainly not consider him so. Odds are he'd either flee to some Christian court, or be assassinated and replaced with a faithful Muslim.
As for your original supposition, the enthronement of a Christian Caliph by Christian invaders would be universally considered invalid. A Caliph is not simply appointed.