The church, which teaches that gay sex is a sin, has expressed disappointment in the recent ruling. Some critics and conservative groups want to overturn it through a constitutional amendment.
In a statement Wednesday, the executive director of Affirmation said he hopes church leaders would not ''use their energies and their funds" to overturn a ruling that affirms the worth of families or meddle in politics that demonizes gays.
''Rather work with us to counsel and assist individuals and families who have been devastated by the church's teachings that have caused family members to reject their children who are gay," Olin Thomas wrote.
''We encourage LDS leaders to find a new focus by preaching and living a Gospel of love and respect toward all peoples and all families," he said of The Church of Jesus Christ of Latter-day Saints.
Thomas said the California ruling doesn't affect Mormon church doctrine or the sacred ceremonies conducted in Latter-day Saint temples, which bind couples for eternity.
Same-sex couples will be able to wed in California beginning June 17.
''This is about civil marriage as an institution of the state - a legal institution based on core societal values, such as the worth and dignity of the family unit," Thomas said. ''Religions are not being asked to marry anyone they don't want to marry."
Kim Farah, a spokeswoman for the 13-million-member denomination, said church leaders had no comment Wednesday.
The Salt Lake City-based church was part of a coalition, including Roman Catholics, evangelicals and Orthodox Jews, which filed court papers in the California case asking the high court to uphold a ban on gay marriage.
Mormons believe God only sanctions marriage between a man and a woman and tout the traditional family as the basic unit of society. Gays are welcomed as church members and can hold church roles, but only if they remain celibate.
Some who have acted on what the church calls ''same-gender attraction" have been excommunicated.
Formed by students at church-owned Brigham Young University in the late 1970s, Affirmation has more than 2,000 gay, lesbian and transgender members. It is not sanctioned by the church and until recently was not even acknowledged by leaders.
In its May 15 statement, the church said it recognizes ''that same-sex marriage can be an emotional and divisive issue" and called the court's ruling ''unfortunate."
It's unclear whether the church will play a role in trying to place a constitutional amendment on California's fall ballot. But Mormons have a history of fighting gay marriage initiatives nationwide.
In 2000, the church actively supported Proposition 22, a ballot measure that prohibited California from legally recognizing gay marriages performed outside the state.
In a letter read from the pulpit, the 740,000 Mormons in California were urged to give time and money to get the measure passed. It was approved but struck down by the recent court ruling.
In February, Affirmation leaders invited new church President Thomas S. Monson to open a discussion about issues important to gay Mormons and their families.
A meeting with LDS Family Services Commissioner Fred C. Riley is scheduled for Aug. 11, said Dave Melson, Affirmation's assistant executive director.