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Carly Fiorina walked onstage with 10 male counterparts at the Republican presidential debate in Simi Valley, Calif., on Wednesday.Credit Monica Almeida/The New York Times

With a debate performance that was steely and at times deeply personal, Carly Fiorina appears to have improved her standing in the race to be the Republican nominee. But even if she falls short, she took a big stride toward filling a role her party badly needs: a credible antidote to the gender gap and the Democrats’ claims of a Republican “war on women.”

The question, as many Republicans see it, is whether they can seize on the opening. A series of events building on the horizon threatens to feed into efforts by the Democrats to paint the Republican Party as tone-deaf on women’s issues.

Republicans in Congress are threatening a government shutdown in a dispute over funding Planned Parenthood. The government’s spending authority expires in less than two weeks, and many conservatives have threatened to vote against a new budget if it includes any money for the organization.

In a separate move, Senate Republicans are moving quickly to schedule a vote on legislation that would impose a federal ban on abortion at 20 weeks of pregnancy. It would be the most restrictive abortion bill to reach the Senate floor in a decade. Similar laws across the country are facing legal challenges from women’s groups that argue they are unconstitutional.

Continue reading the main story Graphic Trump Remains Center Stage at 2nd G.O.P. Debate Prompts from the moderators and new combativeness from the other candidates ensured that Donald J. Trump once again led in airtime.

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But with Hillary Rodham Clinton’s popularity falling among women as questions about her honesty and integrity continue to shroud her, many Republicans are beginning to see Mrs. Fiorina as someone who, if she is not the presidential nominee, is a much more serious contender for the vice-presidential nomination or a large leadership role of some kind within the party.

“If she’s not on the ticket, they would be foolish not to offer her a key role in the next administration,” said Penny Young Nance, president of Concerned Women for America, a conservative organization. The Republican Party, she said, has begun to correct its past mistakes of neglecting its female talent. “I think they’re learning their lesson.”

At the debate at the Ronald Reagan Presidential Library on Wednesday night, Mrs. Fiorina spoke directly to the women watching in a way that none of the 10 men with her on stage could. And yet she was cautious about embracing the role of the sole female on the stage.

At one point, she was the heartbroken mother. Describing how she lost her stepdaughter to drug abuse, she revealed a painful episode that many people watching were unaware of: “My husband Frank and I buried a child to drug addiction.”

She said she knew how disgusted women everywhere must have felt when Mr. Trump said — and then later denied saying — that she was too ugly to be president. “I think women all over this country heard very clearly what Mr. Trump said,” she responded.

And as some of the men around her struggled to name a woman who was not an immediate family member to put on the $10 bill, Mrs. Fiorina dismissed the whole exercise as empty symbolism. “I think, honestly, it’s a gesture. I don’t think it helps to change our history,” she said. “We ought to recognize that women are not a special interest group.”

Her answer on the $10 bill question, said Kristen Soltis Anderson, a Republican pollster and author, was instructive in how it revealed Mrs. Fiorina’s discomfort with being seen as someone trying to capitalize on her gender. “She resists that,” Ms. Anderson said, noting that her own research has shown younger women do not find it as important to vote for the first female president as older women do.

“It’s older women who are more likely to confess, ‘Yeah, I give Hillary Clinton bonus points because she’s a woman,’ ” Ms. Anderson added, “whereas younger woman say, ‘I know there’s going to be a woman president in my lifetime, so we should just focus on who the most qualified candidate is.’ ”

Winning the female vote is challenging for any Republican. Women have traditionally identified strongly with the Democratic Party, while male voters are split between the two parties.

The largest gender gap ever was recorded in the 2012 presidential election: 20 percentage points, according to Gallup. Mitt Romney won among men by 8 percentage points while President Obama won women by 12 percentage points over all. But among white women, Mr. Obama lost with just 42 percent to Mr. Romney’s 56 percent.

To win the White House in 2016, Republicans would almost certainly need to win more than 56 percent of the white female vote. Many see Mrs. Fiorina as a way to help them do that, either as the nominee or in some other role.

“Many Republicans will see her, even if she’s not the nominee, as that magic key that can unlock the gender gap,” said Bruce Haynes, president of Purple Strategies, a Republican consulting firm. “That’s a challenge that many have feared has been set in concrete over time.”

But building a credible campaign, Mr. Haynes added, is far more difficult than having a strong debate performance. And keeping that credibility intact could get much harder in a government shutdown. “You don’t control the agenda, and you have to react to the agenda of others — such as congressional Republicans who want to back everybody up against a wall as usual,” he said.

For her part, Mrs. Fiorina is siding with the hard-liners in her party who want to see a veto showdown with Mr. Obama, who has said he will reject any attempt to take away Planned Parenthood’s funding.

“This is about the character of our nation,” she said emphatically at the debate, receiving some of the most vigorous applause of the night. “And if we will not stand up and force President Obama to veto this bill, shame on us.”

Democrats said that voters should not be confused about what Mrs. Fiorina is really saying.

“You do that and the government shuts down. Plain and simple,” said Representative Debbie Wasserman Schultz, chairwoman of the Democratic National Committee. “She is indistinguishable, other than her gender, from her counterparts on that stage.”

Democrats quickly exposed a flaw in Mrs. Fiorina’s debate comments about the undercover videos of Planned Parenthood that have led to the calls to eliminate its funding. She dared the president and Mrs. Clinton to watch the tapes, saying, “Watch a fully formed fetus on the table, its heart beating, its legs kicking while someone says we have to keep it alive to harvest its brain.”

The video that Mrs. Fiorina seems to be referring to does show a still image of a fetus being held outside the womb. But it is not seen squirming as Mrs. Fiorina describes, nor is there any indication that it is about to have its brain removed.

With her social and fiscal conservatism, Mrs. Fiorina would seem to be a good fit for a Republican nominating contest. A general election, however, is something some Democrats said they would welcome.

“She was Mitt Romney before Mitt Romney was Mitt Romney,” said Mark Mellman, the Democratic pollster who worked on the 2010 Senate race in which Senator Barbara Boxer defeated Mrs. Fiorina by 10 percentage points, in large part by highlighting how as chief executive of Hewlett-Packard she had cut tens of thousands of jobs.

Mrs. Fiorina not only lost that race 52 to 42 percent, she lost women too, getting only 39 percent of their votes, according to exit polls.

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