BOCA RATON, Fla. — Mitt Romney crossed a major threshold early this week, briefly moving above 50 percent in his favorability rating, according to the Real Clear Politics average of polls — giving the Republican a lead over President Obama for the first time on that measure.

Now, with the presidential debates behind him, Mr. Romney has punctured Mr. Obama’s effort to make him unacceptable in the minds of voters, and they enter the final two-week stretch of the campaign having once again turned the election into a referendum on the president.

“The debates — especially the first one — destroyed the Obama crew’s strategy of disqualification,” said Republican pollster Mike McKenna. “Six months of work and $400 million of ad buys went up in smoke in about 10 days. With less than 340 hours to go, they are having real trouble with their footing.”

The debates’ effect can be seen in the favorability ratings. At the end of September, ahead of the debates, Mr. Romney had a 44.5 percent favorable rating. But by Monday, when he and Mr. Obama faced off for the third and final time, the Republican’s rating had leapt to 50.5 percent.

Tom Jensen, director of Public Policy Polling, a Democratic firm, said Mr. Romney’s favorability surge “really has been remarkable” and explains why Mr. Obama has not been able to put away the race at this point.

“It was inevitable that Republicans were going to warm up to him once he became their nominee, but ever since his big victory in the first debate his numbers with independents have improved a good deal as well,” he said. “We’re actually finding in our national tracking now that Romney’s favorability numbers are better than Obama‘s, which no one could have imagined six months ago.”

By Tuesday, Mr. Romney’s favorability average at Real Clear Politics had dropped below 50 percent again, though he still leads Mr. Obama when it comes to net favorability — the calculation of favorable rating minus unfavorable rating.

Mr. Obama’s favorability averaged 49.7 percent, or 4.5 points more than his unfavorable rating. Mr. Romney’s favorability was 49.3 percent, or 6.5 points more than his unfavorable rating.

Campaigning in Henderson, Nev., on Tuesday, Mr. Romney said the debates “supercharged” his supporters.

Less than 24 hours earlier, he had delivered a measured performance in a debate that focused on foreign policy.

While pundits complained that he didn’t leave much daylight between himself and Mr. Obama on the issues, depriving them of the chance to compare and contrast policies, Republicans said Mr. Romney accomplished something deeper — he made himself a palatable alternative to Mr. Obama.

At one point, he even swatted away an Obama attack by accusing the president of offering little else.

“Attacking me is not an agenda,” the Republican said.

He hewed closely to the president’s decision-making when it came to action in Syria, Iran and Afghanistan.

Indeed, at one point the president told Mr. Romney that the only difference in their positions was “you’d say them louder.”

But on domestic issues, where all sides agree the election will be decided, Mr. Obama said he has used the debates to give voters a framework to choose.

“You know, over the last four years, we’ve made real progress digging our way out of policies that gave us two prolonged wars, record deficits and the worst economic crisis since the Great Depression,” he said at the end of Monday’s debate. “And Gov. Romney wants to take us back to those policies: a foreign policy that’s wrong and reckless; economic policies that won’t create jobs, won’t reduce our deficit, but will make sure that folks at the very top don’t have to play by the same rules that you do.”

Instant polls showed Mr. Obama won the debate on points, and commentators on both sides of the aisle skewered Mr. Romney for failing to give a sense of what he would do differently on world hot spots.

“In fact, Gov. Romney appeared to leave a lot of his positions behind, and it does leave you with the question: What is his worldview? What does he really believe?” said Nicholas Burns, a top State Department official in the George W. Bush administration, speaking to CNN on Tuesday. “I think he’s leaving the impression that he’s not quite sure what he’d do or that he’s not being as specific as he might be.”

The Obama campaign said Mr. Romney “proved yet again that he would say anything to close the deal, no matter what his real positions are.”

But Republicans said the debate was a win because Mr. Romney again appeared measured and presidential — the threshold he needs to cross in voters’ minds in order to be a credible alternative.

During October, he also connected with voters in a way he never had before.

A day ahead of the first debate, Comedy Central’s election page was able to write a headline poking fun of the candidate’s appeal: “Romney continues to keep his humanity a closely guarded secret.”

But by the time the first debate was finished, a huge national audience saw Mr. Romney sprinkle in stories of everyday voters he had met who were struggling through the sluggish economy. The Republican’s humanity became the storyline.

Mr. Romney then took that strategy on the road, adding into his standard stump speech a litany of personal interactions, including with a woman whose husband, an Army sniper, was killed in Afghanistan, and the Boy Scout troop that sent its American flag on the Space Shuttle Challenger on its fateful last flight in 1986.

John Zogby, a pollster for The Washington Times, said Mr. Romney’s favorability surge is a significant development in the race.

“Voters got to see an option. Now that there is an option, Romney is viewed as favorably as Obama as a person,” he said.

That’s not to say Mr. Romney has no problems when it comes to relating.

In The Washington Times/Zogby Poll released over the weekend, Mr. Romney trails Mr. Obama and even his own running mate, Rep. Paul Ryan, when voters were asked who is the most likable person on the Democratic and Republican tickets.

Indeed, Mr. Obama nearly doubled Mr. Romney’s rating, 40 percent to 22 percent. Mr. Ryan was rated most likable by 23 percent, and 10 percent said that honor went to Vice President Joseph R. Biden.

The problem extends to the candidates’ wives, too. The Times/Zogby Poll found that voters preferred Michelle Obama to be first lady over Ann Romney, 36 percent to 26 percent. Another 17 percent said both were equally suited, while the rest weren’t sure either made a good first lady.

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