The debate over the freedom to marry is solely about the right of gay and lesbian couples to enter into the government-created institution of civil marriage. Civil marriage consists of two, unrelated adults entering into a legal contract which is recognized by the government. As a result of this contract, the adult couple is granted certain rights and assigned certain responsibilities. Additionally, there is a commonly understood societal status associated with being a married couple. Marriage is a civil right and all loving and committed, consenting, unrelated adult couples should be allowed to enter into the institution if they so choose.
When clergy or congregations of religious faith communities marry couples it is a religious rite/ceremony, not a civil contract. Clergy and congregations choose whom they marry. They are not compelled or required to accept the government's definition of marriage. Some religious institutions are more restrictive than the state, rejecting interfaith marriages or re-marriages after divorce. Those with broader definitions, bless the unions of same-gender couples.
In the U.S., a marriage is only legal with the signing of a civil marriage license. Many couples get married by a judge, a justice of the peace, or other public officiant. They do not need to go to a church, synagogue or mosque in order to marry.
Our government has made the process simpler by allowing religious leaders to perform a religious wedding and to act as a civil officiant. The religious leader must sign the civil marriage 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.
Even after civil marriage became available to all same-sex couples in all 50 states, religions continued to retain the right to decide for themselves whether to perform or recognize any marriage, just as they did before. 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 each faith to decide which marriages it will perform and embrace.
Here is an excellent article entitled, A Conservative Christian Case for Civil Same-Sex Marriage that refutes conservative Christians' belief that civil marriage for same-sex couples should not be recognized.