Which denomination allows homosexuality and what Biblical support do they have to accept homosexuals in the church?
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As mentioned by @ChrisSunami above, the Episcopal Church is the largest. There is also the Evangelical Lutherans, the United Churches of Christ, and the Metropolitan Community Churches. Then there are individual congregations that affirm LGBTs within the United Methodist Church, the Brethren, the Quakers, the American Baptists, and the Presbyterians. Then there are plenty of non-denominational and independent churches that are LGBT affirming (gaychurch.org has a list).
The Biblical defense of their position is usually that we either misinterpret the 6 passages that seem to condemn homosexuality (there is only 6 that seem to condemn it, whereas every other sin seems to be condemned hundreds of times), or the verses or mistranslations.
The story of Sodom and Gomorrah (Genesis 19) is about gang rape, not about monogamous same sex relationships. The 2 verses in Leviticus (Leviticus 18:22 and 20:13) seem to be in the context of Molech worship, which would indicate that what is being condemned is ritual same-sex sex, again not monogamous same sex relationships. The Romans passage (Romans 1:26-27) is describing idolatry leading to same-sex sexual rituals, not homosexuality in general (read the entire chapter, not just verses 26-27). Then we have the problem of translating the terms arsenkoitai and malakoi in 1 Corinthians 6:9 and 1 Timothy 1:10. No one really knows what arsenkoitai means because Paul coined the term. Many LGBT-affirming Christians believe this is harking back to the Leviticus passages and thus refers to those participating in the same-sex sexual rituals.
If you accept these interpretations, monogamous same-sex relationships are never condemned in Scripture.
The Episcopal Church in the United States is probably the largest and most prominent Christian Protestant denomination to publicly affirm support for full inclusion of gay and lesbian members, notably electing the openly gay and presumably non-celibate Gene Robinson to the post of Bishop of New Hampshire in 2003. I'm not aware of the specifics underlying their decision, but I believe Robinson has written extensively on the subject.