MOUNT VERNON, Iowa – Picking up where their contentious debate left off, President Obama and challenger Mitt Romney battled Wednesday for the support of female voters, underscoring their potentially decisive role in settling the fiercely competitive race.

Buoyed by a much-improved performance Tuesday night, Obama traveled to the swing state of Iowa, where he renewed his attacks on Romney for proposing an end to federal funding for Planned Parenthood, and again touted legislation he signed making it easier for women to sue for job discrimination.

“When Gov. Romney was asked about it, his campaign said, ‘We’ll get back to you,’” Obama said of the legislation, repeating a line from the debate. “That shouldn’t be a complicated question: Equal pay for equal work.”

Romney stumped in Virginia, another battleground, where he suggested women had borne the brunt of hardship during an Obama tenure marked by economic anxiety.

“Why is it that there are 3.6 million more women in poverty today than when the president took office?” Romney demanded during a stop at Tidewater Community College in Chesapeake. “This president has failed America’s women. They’ve suffered.”

Women have been a key constituency for Obama, and their enthusiastic backing is vital to his re-election hopes. The president has counted on a strong showing among women to offset Romney’s edge among men. Generally, Obama has been strongest among younger and single women, while Romney has been most popular among older and married women.

After Romney’s commanding debate performance two weeks ago in Denver, polls found many women giving the Massachusetts governor a second look.

“People, especially women, have heard all this negative advertisement against Gov. Romney,” said Rich Beeson, political director of his campaign. “They saw Gov. Romney in the debate and saw an unfiltered view of his plans — what he would do — and I think it resonated.”

That accounted for some of the gains Romney had made in opinion polling, which encouraged Republicans and prompted Obama and his Democratic allies to redouble their courtship of women.

Even as Romney focused his remarks Wednesday on the economy, his campaign launched a new TV spot that sought to reassure women — especially more moderate women — about his positions on contraception and abortion.

In the ad, a woman states her concern that Romney opposes all abortions as well as contraception, but says that after research she learned he does not oppose contraception “at all” and allows for abortion in the cases of rape, incest or to spare the life of the mother.

“I’m more concerned about the debt our children will be left with,” she concludes. “I voted for President Obama last time. We just can’t afford four more years.”

The ad marked a significant departure for Romney, not least because the ad refers to abortion as a form of contraception, a notion that infuriates evangelical and social conservatives, whom Romney heavily courted during the primary season.

As it began airing, a USA Today/Gallup poll was released showing that in 12 key swing states, women named abortion as their most important issue. It was twice as important to them as jobs, which was the prime issue cited by men.

Democrats accused Romney of seeking to whitewash the positions he took during his nominating fight — including support for overturning Roe vs. Wade, the Supreme Court decision that legalized abortion.

In the past, Romney also has supported a measure that would allow any employer to refuse to include contraception coverage in their insurance plans.

“I travel around the country and I talk to women,” said Cecile Richards, who is on leave from her job as president of the Planned Parenthood Federation of America as she campaigns for Obama. “The thought that their daughters or granddaughters would live in a post-Roe world is unthinkable.”

The Obama campaign has hit back at Romney with a commercial highlighting his vow to eliminate federal funding for Planned Parenthood, a position the president brought up repeatedly in Tuesday night’s debate. Similar ads are in the works, according to campaign insiders.

For some women — especially those who are undecided or still open to changing their minds — the debates have served as sort of introduction to the campaign, increasing the stakes for both candidates.

“Women with families are especially busy,” said Robert Durant, a professor of public policy at American University. When it comes to campaigns, “they arrive at the theater midway through the third act, look around, and decide who the heroes and villains are,” Durant said.

During the second debate, the questions and answers dovetailed with that heightened attention. Obama repeatedly sought to frame his support for equal pay, and for contraceptive and health care coverage, as both moral and economic matters important to women.

Romney accented his support as governor for flexible schedules for his aides, but did not answer a question from the audience — and a challenge from Obama — about whether he supported equal pay for women.

The two candidates continued to spar Wednesday from afar, picking up on points both sought to make the night before.

Appearing at Iowa’s Cornell College, and wearing a pink breast-cancer awareness bracelet, Obama tweaked Romney for saying that as Massachusetts governor he brought gender balance to his cabinet by leafing through “binders full of women.”

Pitching careers in science, technology, engineering and math, Obama said, “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.”

“And,” he added, “when young women graduate, they should get equal pay for equal work.”

The “binders full of women” remark became an Internet and Twitter sensation immediately after the debate, spawning countless parodies, websites and even music videos.

Spirit Airlines jumped on the phenomenon by launching a “Binders Full of Sales” promotion. The Republican National Committee responded to the jokes with an email depicting an empty binder, which it described as Obama’s agenda for the next four years.

In Chesapeake, Romney cast the economic argument far differently.

As he has campaigned across the nation, Romney said, he routinely hears from women who want better jobs for themselves, their spouses and their children, as well as better schools.

“That’s what the women of America are concerned about, and the answers are coming from us, not from Barack Obama,” he said.