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Phyllis Schlafly, president of the Eagle Forum, responded by calling on Republican National Committee Chairman Reince Priebus to reinforce the party’s commitment to a federal marriage amendment and DOMA.

Weeks later. the RNC released a post-election analysis — called the “Growth and Opportunity Project” — that found there is a “generational difference within the conservative movement about issues involving the treatment and rights of gays — and for many younger voters, these issues are a gateway into whether the party is a place they want to be.”

“We need to campaign among Hispanic, black, Asian and gay Americans and demonstrate that we care about them, too,” the report said.

Jimmy LaSalvia, co-chairman of GOProud, which represents gay conservatives, predicted that more candidates for the nation’s top office will come down on the side of states’ rights on the marriage issue.

“Those advertising the support of the federal marriage amendment look completely out of touch because everyone knows that it is very unlikely that that it would ever pass,” Mr. LaSalvia said. “There is a new political reality now. People will vote for candidates regardless of their support on this issue. There is going to come a day when the Republican Party — the vast majority of Republicans — will support same-sex marriage.”

Polls show that public opinion has shifted in favor of same-sex marriage over the past decade and is most popular among the same young voters who overwhelmingly supported Mr. Obama in the 2012 election.

Social conservatives and evangelical Christian leaders, though, argue the polls are “skewed” and that GOP leaders are throwing social conservatives overboard.

Gary Bauer, president of American Values, said Republicans on Capitol Hill “seem afraid to make arguments about any values issues” and said the GOP must take stronger public stands against redefining marriage and abortion — otherwise they will risk losing their base of evangelical and religious supporters.

“I would just say to Republicans generally who want to be president that we are living in an America where liberal courts are day-by-day ripping out of the public square our Judeo-Christian world view,” Mr. Bauer said. “To say that [marriage] ought to be defined state-by-state begs the question: How do you propose we get there? The court is being asked to strike down every state law.”