For the most part, yes, there truly is separation.
As for the Bible references to citizens, that's a different issue. People have the freedom to base their politics on whatever grounds they wish...the state can only intervene on peoples actions, not on their opinions (Reynolds v. the United States (1878)). Those people's biblical quotes are politically irrelevant. We don't argue about biblical interpretations when we make our laws.
As the Prop 8 court case showed, the U.S. only focuses on whether or not there is a national interest served in permitting only heterosexual marriages. As the court ruled, there is no such interest, no matter what supporters say. As such, the judge down the ban in California as unconstitutional and the parties have requested that the federal courts examine the issue. I'm sure that, no matter what they decide, it will be appealed again. This will no doubt eventually be a Supreme Court case.
If there was no separation in the first place, there would be no judicial procedure; lawmakers could whip out their bibles, argue what it says, and write the laws accordingly. That isn't what's happening.
Culturally, there isn't always separation, but, legally, it's there.