Millions of Hispanic voters seconded that emotion Tuesday with resounding 71 percent support for Obama, tightening Democrats’ grip on the White House and putting Republicans on notice that they must seriously court the nation’s largest minority group if they want to win the presidency again.

According to initial exit polls, Republican candidate Mitt Romney, who backed hard-line immigration measures, came away with 27 percent Hispanic support. That’s less than any presidential candidate in 16 years and a sharp drop from the 44 percent claimed by President George W. Bush in 2004 after he embraced immigration reform.

“We could have won this election if the party had a better brand name with Hispanics,” said Al Cardenas, chairman of the American Conservative Union. “I don’t believe there’s a path to the White House in the future that doesn’t include 38 percent-40 percent Hispanic support.”

Cardenas said Hispanics were only a large part of a worrisome trend in the electorate, which is increasingly comprised of younger and minority voters who traditionally do not back Republicans. If the 1980 electorate looked like the 2012 version, he added, Jimmy Carter would have defeated Ronald Reagan.

Matt Schlapp, who was political director of George W. Bush’s 2000 campaign, drew parallels between the Republicans’ standing with Hispanics and the party’s troubles with African-Americans, who now routinely back Democrats by 9-1 margins. “The idea that we would somehow copy that with the Hispanic community is troubling,” he said.

Hispanics have long favored Democrats, but they have been trending even more sharply toward that party since Republicans stymied Bush’s immigration proposal and favored hard-line immigration measures that critics decried as racially motivated.

Romney chose an author of Arizona’s controversial immigration law to advise him during the Republican primaries and called for “self-deportation” to lower the number of illegal immigrants. Obama, meanwhile, announced in June that immigration authorities would grant work permits to people brought here illegally as children who graduated high school or served in the military.

The directive energized a Hispanic electorate that had been disappointed by Obama’s inability to pass immigration reforms.

A survey of Hispanic voters by the firm Latino Decisions found that Hispanics gave Obama his winning margin in Colorado, Florida and Nevada, swing states where they turned out in unusually high numbers. Even before the races were called, some Republicans took to the airwaves and social media calling for the party to back off its hard-line stance and embrace certain immigration reforms.

It’s unclear whether the results would change the Republican Party’s opposition to legalizing some illegal immigrants. In a conversation with the Des Moines Register newspaper last month, Obama predicted that Republican opposition could crumble after Hispanics delivered the White House to him. The conversation was initially off the record but later published with the president’s consent.

“And since this is off the record, I will just be very blunt,” Obama said. “Should I win a second term, a big reason I will win a second term is because the Republican nominee and the Republican Party have so alienated the fastest-growing demographic group in the country, the Latino community.”

On Wednesday, Senate Majority Leader Harry Reid, a Democrat, vowed to introduce an immigration reform package next year, saying Republicans would reject it “at their peril.”

Opponents of an immigration deal warned that Republicans should not take the Democrats’ bait. Steve Camarota of the Center for Immigration Studies noted that Hispanics have reliably backed Democrats for decades, even after President Ronald Reagan signed an immigration amnesty in 1986 that gave many of them legal status.

Those new American citizens, Camarota said, turned into Democrats.

Camarota and other supporters of immigration restrictions contend that Hispanics lean Democratic because they favor government social programs and higher taxes on the wealthy. “They changed the national electorate, and now they have to move with the electorate,” Camarota said of the Republicans. “For 30 years that we have data, Hispanics have been voting Democratic. There’s no reason to think that’s going to change unless the Republican Party moves away from its low-tax, low-regulation position.”

NumbersUSA President Roy Beck, whose group advocates reductions in immigration levels, argues that Republicans like Romney need to explain to Hispanic voters why immigration restrictions are in their interest. “He should have talked about Hispanic unemployment and how much high immigration hurts Hispanic employment.”

The debate is nothing new for the Republicans.

Mario H. Lopez, president of the conservative Hispanic Leadership Fund, said he’s heard arguments like that before — after every election in which Hispanics lean more Democratic and Republicans suffer. “The clock has been ticking,” Lopez said. “Some of us have been talking about it for years. It’s up to them if they want to listen or have more nights like Tuesday night.”