WASHINGTON (AP) -- Republican lawmakers have a message for those who want the party to soften its emphasis on social conservatism in hopes of reaching a wider national audience: Not so fast.

House Republicans flexed their cultural and conservative muscles Tuesday, passing the most restrictive abortion measure in years. They also advanced legislation to crack down on immigrants living illegally in the country, even as senators pursue a plan that would offer those same millions a shot at citizenship.

The actions reflect a roiling debate among Republicans over why they lost two elections to President Barack Obama, and how best to rebuild a winning formula.

Many Republicans in Congress and elsewhere think the party's establishment erred in concluding the GOP must embrace "comprehensive immigration reform" to attract Hispanic voters. And they dismiss the notion that Republicans should soft-pedal their opposition to abortion, a subject on which they say public opinion is moving their way.

"There's been a misleading thought as to what happened after the last election cycle," said Rep. John Fleming, R-La.

"Most Americans do not support amnesty, especially without securing the borders," he said, regarding the idea of citizenship for those here illegally. As for abortion, Fleming said, there's growing public concern about second-trimester abortions, "so we're actually gaining ground."

Like Democrats, Republicans often discuss ways to keep their base loyal while attracting independent voters near the political center. The urgency rose last fall, when Mitt Romney became the fifth Republican in six presidential elections to lose the popular vote.

On abortion and reproductive rights, some strategists say the greatest need is for Republicans — especially men — to steer clear of incendiary language such as "legitimate rape." They know there's no way the Democratic-led Senate will embrace the House bill, which would bar abortions 20 weeks after conception.

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Tuesday's debate was largely symbolic but important, Republican leaders said.

Immigration's fate in Congress is less certain. It's increasingly clear, however, that many Republicans think party elders were hasty in saying the GOP won't win future presidential elections unless it agrees to far-reaching immigration changes that include new pathways to citizenship.

"What an idiot," Washington state Republican Chairman Kirby Wilbur said Tuesday of Sen. Lindsey Graham's recent comments on the matter. Graham, R-S.C., said that without "immigration reform" along the lines the Senate is weighing, "we're in a demographic death spiral as a party."

"The pathology report of the death of the Republican Party is grossly overstated," Wilbur said. Republicans must do better jobs of messaging and finding voters, but they should not overreact to Romney's relatively narrow loss to Obama, he said. Obama won 51 percent of the popular vote to Romney's 47 percent but defeated the Republican by a wide margin in the Electoral College, 332-206.

Democrats want to portray Republicans as out of step with the nation's values on gay rights, women's rights and common-sense solutions to illegal immigration.

During Tuesday's House debate on abortion, Rep. Ami Bera, D-Calif., asked to bring up a student loan bill instead. "This is a direct attack on women's rights," he said after being overruled.

Republicans responded by sending a parade of women to the House microphones.

"We are changing hearts and minds," said Rep. Ann Wagner, R-Mo. "We hear more and more evidence that life begins at conception." She said she covets the day when abortion is "absolutely unthinkable."

Rep. Jim Gerlach, R-Pa., said he supports the anti-abortion bill, but "our focus really ought to be on jobs, the economy and those kinds of issues."

But Fleming, a former Navy and family physician, said there's no political or policy harm in trying to restrict abortion.

And the best way to win over Hispanics, blacks and Asians, he said, is to encourage the conservatives among them to enter local politics and run for office as Republicans.

"I see a bright, bright future for the Republican Party and conservatism in general," Fleming said. But it will happen only if "we have people of all sectors who join us in our beliefs and principles," he said.

Recent polls provide fodder for both sides in the immigration debate. They show Americans support allowing those in the country illegally to stay and become citizens if they meet certain requirements. Polls that ask about giving immigrants a chance to become citizens without specifying conditions find less support.

Republicans appear to be following the immigration debate more closely than Democrats. That might help conservative Republicans generate enthusiasm in next year's midterm elections, which typically draw fewer voters than do presidential races.

It's too early to guess how immigration might play in the 2016 presidential contest.

On abortion, polls find little change in Americans' sentiment. The General Social Survey, which has tracked opinion since the 1970s, finds support for legal abortion has been roughly stable. About 4 in 10 last year said abortion should be legal if a woman wants one for any reason, on par with the average results over time.

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Associated Press Director of Polling Jennifer Agiesta contributed to this report.