The Crux of the Issue

From Marriage Equality USA:
The debate over the freedom to marry is about the right to enter into the state-created institution of civil marriage only. After all, marriage is a civil right and two consenting adults should be allowed to enter into the institution if they so choose.
Unlike some religious definitions, civil definitions of marriage do not usually mention childbearing, sexual relations, living arrangements, or religious belief/observance.
When clergy or congregations marry couples it is a religious rite, not a civil ceremony, though the government may recognize it. Clergy and congregations choose whom they marry. They aren't compelled to accept the state's marriage definition, and indeed, many religious institutions don't accept it. Many religious institutions are more restrictive than the state, rejecting interfaith marriages or remarriages after divorce. Some have a broader definition, blessing the unions of same-gender couples. (Marriage Equality USA would like to thank Interfaith Working Group for this passage.)

In the U.S., a marriage is only legal with the signing of a marriage license. That is why many opposite-sex couples can go to a judge or any other public officiant and need not go to a church, synagogue or mosque. Our government has made the process simpler by allowing religious leaders to perform a religious wedding AND to act as a civil officiant. Each religious leader must sign the civil license before witnesses and the couple for the marriage to be legal. In Europe, couples MUST go before a public official to marry. A religious ceremony is 'secondary' and optional -- only occurring if the couple wishes to have one.
This fact is important to note because many same-sex couples are simply interested in the government's acknowledgement of their relationship. We are not asking for any religion to accept our marriages, although, many religious institutions throughout our country do.

• Civil and religious marriage are not the same thing. Many religious faiths already recognize religious unions or marriages between same-sex couples, even though such unions are not recognized by the government.
• Individual congregations of Reform Jews, American Baptists, Buddhists, Episcopalians, Presbyterians, Unitarian Universalists, Methodists, the Society of Friends and the United Church of Christ have performed marriages for same-sex couples.
• Even after civil marriage becomes available to same-sex couples, religions will retain the right to decide for themselves whether to perform or recognize any marriage, just as they already do. No court decision or legislative enactment can change the basic tenets of religious faith. For example, some religions will not marry someone who has already been divorced, although the person is free to marry civilly. We respect the right of a faith to decide for itself what marriages it will embrace.