Republicans seem to care less about civilization than "family", more precisely "conserving [Christian] family values". That's why it's linked to all the nonsensical, irrational thinking about gay marriage and abortion; they think it violates the ideals of the traditional Christian family. It cannot be equated with "civilization" unless you somehow believe that those things are uncivilized, which makes the entire axis analysis tautological.
There are certainly Republicans who are all for family values for other reasons besides Christian morals. They equate this with civilization not in contrast with uncivilization, the question for them isn't "civilization vs. savagery" exactly, it's more nearly "civilization vs. decadence." Open homosexuality and abortion call to mind the Roman Empire in its decline, not the Dark Ages.
A really powerful argument in this vein comes from the Nixon tapes. It's very much worth listening to, no matter what your views on the matter are, because I think it really crystallizes why conservatism and the religious right have so much common ground.
