I'm a lifelong Christian and I've been married a long time, enjoying quite a long list of legal, social, and economic privileges that go with being married.
The Golden Rule--one of the most basic moral principles taught by Christ--says that we must treat others as we ourselves wish to be treated. I would not like to have all those privileges denied me and my spouse because of our respective genders. I conclude I cannot morally support denying those privileges to a same-sex couple on such a basis.
The thing many Christians seem to be missing is that someone else's marriage is about the married couple, not about the rest of us. We've got the freedom to pick someone else to love, and if that person agrees to commit to the relationship, to make that person our official next of kin. It's been over 50 years since the U.S. Supreme Court declared that right couldn't be restricted because of the respective races of the couple. A similar rule about their respective genders makes equal sense.
Churches remain free to refuse to solemnize same-sex marriages, just as they've always been free to refuse to conduct marriage ceremonies for any other reason they choose (including mixed-race marriages). And I remain free to stay away from churches that practice such discrimination.
Maximizing everyone's freedom is the best way to let everyone make their own moral choices.