I think you're being very selective in what you recognize as being Republican values. They care very much who marries whom, and of what value is a marriage if it isn't legally valid? (Remember DOMA?)
What the GOP has managed to get passed as legislature has no reflection on what they would pass if they could. You don't think they wouldn't pass an anti-defamation flag law if they could get it through Congress? An anti-gay law? They're still trying to limit access to abortions at the state level, even though the Supreme Court has ruled women have a right to an abortion if they choose. I suspect, although in fairness I have no proof, that they would also outlaw Islam if given the chance.
But yes, both parties are very interested in what we put in our bodies, but you stated that it was only Democrats doing so. I think there's ample evidence against both sides.