White churches in the south weren't forced to conduct inter-racial marriages in the South in the wake of Loving v Virginia. The Catholic Church hasn't been forced to recognize civil divorce. The Mormon Church hasn't been forced to recognize civil marriages. Hell, it was only 15 years ago that the favorite hang out of Republican nominees stopped formally censuring students in mixed-race relationships. Anti-bigotry laws have not impacted religious institutions and there's no reason to think they will in the future (bakeries aren't religious institutions).
Anyway, the 14th amendment provides the basis of the ruling and there's considerable precedent supporting ssm laid out in the majority ruling. Tying together Loving, Lawrence, Windsor, etc. with the equal protection clause is far more solid than the arguments made in the dissents which were basically about nothing.