WASHINGTON (AP) A dispute over gay marriage between former U.S. Vice President Dick Cheney's daughters is shining a bright light on the Republican Party's broader dilemma with the emotional issue.

Public support for gay marriage has been rising dramatically. Yet a big majority of Republicans oppose it. And they tend to nominate candidates for Congress and the presidency who are out of step with the shift in public opinion, at least at the national level.

Republicans will confront this issue in congressional and governors' races next year. It could play an even bigger role in the 2016 presidential election. Anyone supporting gay marriage would face tough resistance in Republican primaries, which conservatives dominate.

The more attention the issue gets, the more it might alienate much-needed independent voters.

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"Many of us in the Republican Party would like to see this issue go away," said Mark Graul, a political strategist. Still, he said, relatively small numbers of voters consider it their top priority.

Cheney is trying to limit the political damage to his daughter Liz, who wants to oust Sen. Mike Enzi in next year's Wyoming primary. Liz's sister, Mary Cheney, is married to her long-time companion, Heather Poe.

The two sisters began a highly public quarrel Sunday when Liz Cheney told Fox News Sunday she opposes legalized gay marriage, but the issue should be left to the states. Mary Cheney wrote on Facebook: "'Liz -- this isn't just an issue on which we disagree, you're just wrong -- and on the wrong side of history."

Poe went farther. She wrote that Liz Cheney had always supported the lesbian couple and their two children, and "to have her say she doesn't support our right to marry is offensive."

Dick Cheney and his wife, Lynne, tried to calm things on Monday. They said their daughters love each other, but "Liz has always believed in the traditional definition of marriage."

In 2009, Liz Cheney told MSNBC she would oppose a constitutional amendment to ban same-sex marriage in every state. Then, as now, she said states should decide how to deal with same-sex marriage. Fourteen states and the District of Columbia have legalized it.

In July, 54 percent of Americans said same-sex marriage should be legal, compared with 27 percent in 1996, according to Gallup polling.

Support is much higher among people aged 18 to 29.

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Most of the increase was among Democrats and political independents. In July, 28 percent of Republicans supported legalized gay marriage, compared with 16 percent in 1996. Among Democrats, support is now 73 percent.

The 2012 Republican Party platform reaffirmed "our support for a Constitutional amendment defining marriage as the union of one man and one woman."

Many Republicans, however, consider it a losing battle. Nearly three-quarters of Republicans told Pew Research that recognition of same-sex marriage is inevitable.

Some prominent Republicans say the party can embrace the trend without sacrificing conservative values. "What could be more conservative than support for more freedom and less government?" former Republican national chairman Ken Mehlman, who in 2010 announced he is gay, wrote in The Wall Street Journal last year. "And what freedom is more basic than the right to marry the person you love?"

Many conservative groups, however, remain strongly opposed. If same-sex marriage is legalized, "taxpayers, consumer and businesses would be forced to subsidize homosexual relationships," says the Family Research Council, which is influential in some Republican primaries.

Some Republicans seek a middle ground.

New Jersey Gov. Chris Christie proved this month that a Republican official can thrive in some states, anyway by officially opposing gay marriage but saying he will do nothing to block it. In exit polls following his landslide re-election, only 6 percent of New Jersey voters said same-sex marriage was their top issue.

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Associated Press writer Charles Babington and AP Polling Director Jennifer Agiesta contributed to this report.