Republican nominee Mitt Romney’s awkward comment during the second presidential debate that he had received “binders full of women” as Massachusetts governor when he requested more female job candidates went fully viral Wednesday, drawing snickers from voters but also fueling a broader fight between the two campaigns over the key support of women.

Romney’s remark was just a sliver of the discussion Tuesday night about issues relevant to women, with the candidates tussling over subjects such as contraception and unequal pay. The battle escalated on Wednesday, as President Obama worked to reclaim his advantage among women — and as the Romney campaign returned to its core argument that the Republican is better suited to manage women’s top concern, the economy.

Campaigning in Iowa, Obama ridiculed his opponent.

“I’ve got to tell you, we don’t have to collect a bunch of binders to find qualified, talented, driven young women, ready to learn and teach in these fields right now,” he said.

Romney, at a stop in Chesapeake, Va., revisited a question from the debate about the gender pay gap, saying that answers women want to hear about the economy are coming from him, not Obama.

“This is a presidency that has not helped America’s women, and as I go across the country and ask women, ‘What can I do to help?’ what they speak about day in and day out is, ‘Help me find a good job, or a good job for my spouse,’ ” Romney said. “That’s what the women of America are concerned about. And the answers are coming from us and not from Barack Obama.”

Although the candidates have courted female voters all year, they are renewing their attention to the demographic as polls show the race tightening, and as some surveys indicate that Obama’s once-sizable advantage among women has slipped.

Romney shifted his emphasis Tuesday night on at least one issue relevant to women, asserting that “every woman in America should have access to contraceptives.” He objects to the president’s policy that requires employers to pay for contraception as part of health insurance coverage, an issue important to conservatives who consider it an infringement on the rights of religious institutions. But he did not mention that, and instead focused on the undisputed issue of access, as he appeared to be trying to present a more moderate face in the closing weeks before the election.

The Romney campaign also debuted a new ad this week that tries to soften his image. The spot, called “Sarah,” features a young woman who says Obama’s ads accusing Romney of wanting to ban all abortions and contraception “concerned” her.

“So I looked into it,” she says. “Turns out, Romney doesn’t oppose contraception at all. In fact, he thinks abortion should be an option in cases of rape, incest or to save a mother’s life.”

“This issue is important to me,” she says. “But I’m more concerned about the debt our children could be left with.”

The ad is a direct play for undecided voters such as Paula Fultz, 59, who is from a Cleveland suburb. She supports abortion rights and backed Obama four years ago but gave Romney a closer look after the first debate. “He seemed more credible than what I’d seen before,” Fultz said of Romney. “So I’ve been leaning more to looking at the jobs discussion.” After Tuesday’s debate, Fultz said she is still leaning toward Romney, even though she said she thought Obama performed better the second time around.

The Obama campaign is working to blunt similar shifts among women in swing states. White House senior adviser David Plouffe, traveling to Iowa after the debate, previewed the team’s plan to argue that Romney is on the wrong side of women’s issues, such as his support for the Blunt Amendment, which allows employers not to pay for birth control if they have a moral objection to doing so.

“Mitt Romney: Wrong for American Women,” read a press release from the campaign late Wednesday. During his stops that day, Obama wore a pink bracelet for breast cancer awareness.

In the Tuesday night face-off, Romney said he had a strong record of hiring women, saying one key to doing so for top jobs was allowing family-friendly work hours.

“I recognized that if you’re going to have women in the workforce, that sometimes you need to be more flexible,” he said, recalling that his gubernatorial chief of staff had two school-age children. “She said, ‘I can’t be here until 7 or 8 o’clock at night. I need to be able to get home at 5 o’clock so I can be there for making dinner for my kids and being with them when they get home from school.’ So we said, ‘Fine. Let’s have a flexible schedule so you can have hours that work for you.’ ”

Obama spoke about growing up with a single, working mother and a working grandmother who trained men for jobs that paid more than hers. He also talked about signing the Lilly Ledbetter Fair Pay Act of 2009 — which made it easier for women to file lawsuits alleging pay discrimination — as one of his first actions in office.

Vice President Biden, campaigning in Colorado on Wednesday, picked up on the issue of whether Romney had sought female employees. “You heard the debate last night. When Governor Romney was asked a direct question about equal pay, he started talking about binders,” Biden said. “Whoa! The idea that he had to go and ask where a qualified woman was, he just should have come to my house. He didn’t need a binder.”

Polling about where women stand in the race has been a point of controversy, with seemingly contradictory data pouring in each day.

Two recent surveys — a national Pew poll after the first debate and a more recent USA Today-Gallup survey in 12 battleground states — had Obama and Romney tied among female voters, something that would be a historic shift away from a gender gap that has helped Democrats in recent elections.

A Quinnipiac University poll in Pennsylvania released Tuesday had Romney closing in on Obama there but had the president with an 18-point advantage among women who are likely to vote. But a new poll from Marquette University Law School shows Romney making big gains in Wisconsin, entirely by winning over women.

In the new Washington Post-ABC News national poll, 51 percent of women back Obama and 44 percent support Romney, with the seven-point margin a numerical, but not statistically significant, advantage for the president.

Amy Gardner in Iowa and Ohio and Jon Cohen in Washington contributed to this report.