Obviously, marriage isn't about religious belief, since a large number of nonreligious/agnostic/atheist people are married.

And government administration of marriage records dates back to medieval England (the source of the US common law), so the notion that the government is new to the game is also a bit specious.

But let's examine the idea of marriage as private contract:

1) Private contracts don't apply to taxation and immigration, which are both the exclusive purview of government (and explicitly cite marriage);

2) Any effort to unilaterally end a marriage contract (through nonperformance, dispute, etc.) would automatically "involve the government," since it would result in a stop in court.

"Private contracts" have government involvement as well, for the simple fact that contracts themselves are defined through case law, common law and legislation (such as the Universal Commercial Code) and enforced/nullified through courts.

As for the "Christian nation" stuff, that's a dead horse and largely inaccurate, and wouldn't matter even if it was more accurate, since 18th century Christianity is VERY different from the Joel Osteen new-agey selective literalism school of popular "Christianity" that most Americans generally adhere to, across denominations.

And finally, as a Jew, I'd appreciate it if people would stop citing the fantasy of "Judeo-Christian traditions." Contemporary Judaism is rooted in the belief that Christianity is fundamentally inaccurate as a base premise. It also recognizes that the sections of the Tanach that Christian monarchs in medieval Europe edited into "Leviticus" (and that are most popularly quoted by American Christians today in their anti-gay crusade) are the laws governing Levite rabbis and rabbinical/temple assistants, not the general population (or even the other 11 tribes).

You might as well discuss "Islamo-Christian beliefs."