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 group also said that while women made of 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.