Marriage is not a religious ceremony anymore. Sure it used to be but people get married at court houses without ever seeing the inside of a church all the time. The second the government started giving benefits to those who are married over those who aren't it couldn't be a religious thing. If you want it to go back to being a religious institution then I'd be fine with that, we can just take away all the legal benefits that go along with being married and have those relegated to a "union".
As far as if it's a right to get married, while not explicit we in the US specifically have the right to pursue happiness as long as it doesn't hurt others, and getting married falls right in line with that in my mind. So no there is nothing in the constitution that says "you have the right to get married", but there are general terms that would lead one to think you do.
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