As far as I can tell, Christian fundamentalists are opposed to homosexuality on the basis of verses in Leviticus and Romans.
But those are proscriptions on sexual acts, and nobody is asking the couples in Kentucky what they plan to do when they get home. Does Kim Davis think she's being asked to issue a license for the couples to have sex?
Of course, it is tempting to argue against her on the basis of her misunderstanding of Scripture and her hypocrisy. But those arguments shouldn't even have to be joined; there is no need to play the game on her turf. The First Amendment, to which she appeals, begins with the Establishment Clause. And she took an oath. That is all you need to know.