The battle for women’s votes helped shape Tuesday night’s debate between President Obama and Republican challenger Mitt Romney, pushing issues such as equal pay for women and access to contraception to the forefront of the closely fought race.

Commentators and social media sites seized on Romney’s claim that he had sought and received “binders full of” female job candidates after being elected governor of Massachusetts and learning that his advisers were recommending mostly men for cabinet-level positions.

But while Romney’s supporters praised the anecdote as evidence that their candidate values diversity, his critics denounced him for not addressing the disparity in pay between women and men, and for singling out women executives — and not their male counterparts — as needing flexible work hours in order to take care of their responsibilities at home.

Meanwhile, the Massachusetts Women’s Political Caucus on Wednesday disputed Romney’s assertions that he initiated the search for female job candidates. In a statement, the group said it approached both Romney and his opponent during the gubernatorial campaign seeking a commitment to hiring parity, and presented Romney with top female applicants for each cabinet position after he was elected.

The Romney campaign responded Wednesday that he had worked with the group to find qualified women for top positions.

The group also said that while women made up 42 percent of hires in the Romney administration, the percentage of women in senior appointed positions dropped to 25 percent over the next two years.

Romney emphasized his strong record of hiring women during Tuesday night’s debate, and claimed that he initiated the contact with women’s groups to help him recruit candidates because he was dissatisfied with the pool of men selected by his advisers. He also said one key to hiring women 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,” Romney 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, in contrast, spoke about growing up with a single, working mother and a working grandmother who found herself training men for jobs where they easily out-earned and outranked her. He also talked about signing the Lilly Ledbetter Fair Pay Act as one of his first official acts in office. The measure makes it easier for people to file lawsuits over pay discrimination.

And while the president cited child-care tax credits as a factor in determining whether women can “go out there and earn a living for their family,” he was quick to add: “These are not just women’s issues. These are family issues. These are economic issues. And one of the things that makes us grow as an economy is when everybody participates and women are getting the same fair deal as men are.”

Vice President Biden, interviewed after the debate on CBS, questioned Romney’s assertion that women job candidates could be hard to find, saying he had “never had any problem when there is a job opening having as many women apply as men.”

“The idea you gotta go ask help to find one, I didn’t quite understand what he was talking about,” Biden said. “They’re in abundance.”

Rep. Paul Ryan (R-Wis.), Romney’s running mate, defended Romney in an interview Wednesday morning, saying the former Massachusetts governor “has an exceptional record of hiring women in very prominent positions in his administration.”

With Obama in the White House, Ryan told CBS, “this economy has been terrible for women. Poverty rates for women are at a 17-year high. We need jobs, we need economic growth. And among those who have been hit hardest by this economy are women.”

The focus on women’s issues comes as some polls show a narrowing of what had been a double-digit advantage for Obama among female voters. Recent surveys by the Pew Research Center and USA Today/Gallup had Obama and Romney tied among women, which would be a historic shift away from a gender gap that has helped Democratic candidates win elections. The Obama campaign pushed back on the methodology of the Gallup 12-state poll, however, claiming numbers in state polling still showed women clearly tilting toward the president.

A Quinnipiac poll in Pennsylvania released Tuesday had Romney closing in on Obama among all likely voters but showed an 18-point advantage for the president among women in that group. In the new Washington Post-ABC News national poll, 51 percent of women backed Obama and 44 percent backed Romney. The seven-point margin represents a numerical, but not statistically significant, advantage for Obama, given the sample size of likely voters.

But a new Wisconsin poll from Marquette University Law School showed a dramatic shift of female voters toward Romney after the first debate.

Across the country, especially in swing states such as Virginia, Democrats are emphasizing abortion rights, funding for Planned Parenthood and the Ledbetter legislation — issues on which more female voters seem to favor Obama’s positions. After the first presidential debate, Democrats complained that neither Obama nor moderator Jim Lehrer had raised those topics.

So they were all ears as undecided voter Katherine Fenton stood up at Tuesday’s town-hall-style debate to ask Obama what he would do to “rectify the inequalities in the workplace, specifically regarding females making only 72 percent of what their male counterparts earn.”

After the ensuing discussion, Romney’s phrase “binders full of women,” quickly became an Internet meme — the “Big Bird” of the second debate, so to speak. A Twitter page filled with snarky comments such as, “Injustice. Binders full of women sold for $0.77 at Staples. Binders full of men selling for $1.00.”

A Facebook page received more than 250,000 “likes,” and a Tumblr page drew dozens of photos and captions mocking Romney’s statement, including a recent photo of Secretary of State Hillary Rodham Clinton typing on her mobile device, with the made-up caption: “Romney still uses binders? LOL.” Many of the Tumblr entries played off the definition of “binder” as something that restrains or restricts movement — clearly not what the GOP White House hopeful intended.

In an interview with reporters en route to Iowa after the debate, White House senior adviser David Plouffe said voters can expect to continue to hear about issues affecting women that came up during the debate. Plouffe said an army of advocates for the president will argue that Romney is on the wrong side of the issues. One example they will cite: the Republican’s support for the Blunt Amendment, which allows employers to opt out of health-coverage mandates, such as paying for birth control, if they have a moral objection to doing so.

Plouffe said women remain crucial to the outcome of the election because they make up a majority of undecided voters in every swing state.

On the campaign trail in Chesapeake, Va., Romney referenced the exchange with Fenton, 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. “And help my kid, make sure my children have a bright future, better schools and better job opportunities. That’s what the women of America are concerned about, and the answers are coming from us and not from Barack Obama.”

Obama, in Iowa, picked up the binders theme as well.

“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.”

In the 2008 election, 10 million more women voted than men. Obama beat Republican nominee John McCain by 13 percentage points among women. Yet, in the 2010 mid-term elections, Republicans managed to draw even with Democrats among women, which helped lead to a takeover of the House by Republicans and a narrowing of the Democratic majority in the Senate.

Biden insisted during the interview with CBS that the new polls showing a dead heat among women were not accurate. “We still have a clear lead among women for good reason; you saw it tonight,” Biden said. “You saw when Governor Romney was asked about everything from what he . . . was going to do about everything from Lilly Ledbetter to equal pay. He just obfuscated the issue.”