Are there any Christian Churches or communities that teach same-sex marriage is a right? On what biblical basis do they base their teaching?
A couple of examples will do for an answer.
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Their statement of note:
The 2013 USA National Conference recommends the sacrament of marriage be extended, where legal in the USA, to persons of the same sex/gender. Thus the 2013 USA National Conference recommends a change to the current policy for the USA on the sacrament of marriage;
And the above recommendation was recently approved.
Here is their basis for it (though not explicitly Biblical):
We believe this is where God is directing Community of Christ, to be an inclusive body where everyone is welcome source
This Wikipedia page lays it out pretty well which Christian denominations accept same-sex marriage and just how much they accept it. Some denominations just kind of accept a little, while others split over it.