I don't really believe in a private vs public Christianity; but I do agree with the political aspect of your point.
The Religious Right has a tendency to view "Christianity" as an instrument of culture and civilization (not necessarily in those explicit of terms).
It's not Christianity or the Church that's threatened, it's an imagined Christian Civilization, concerning which the United States is the last great bastion (seeing Europe as having fallen). Gay marriage is seen as a threat against this imagined Christian culture and civilization.
It boils down more to power than religion, usually.
-CryptoLutheran