Republican presidential candidates on Sunday sought to play up their appeal to women voters, in an effort to counter concern that recent comments from GOP front-runner Donald Trump would hinder the eventual party nominee’s ability to win over women in the general election.

The candidates spoke as Mr. Trump, a real-estate magnate denied his remarks about Fox News host Megyn Kelly were a reference to her menstrual cycle.

Mr. Trump on Friday lashed out at Ms. Kelly, who had asked him in Thursday’s debate about his history of making incendiary remarks about women. He told CNN, “You could see there was blood coming out of her eyes. Blood coming out of her wherever.” Mr. Trump says he was employing a metaphor about her speaking in anger with blood coming from her nose.

On Sunday, former Hewlett-Packard Co. chief executive officer Carly Fiorina, the only woman in the Republican field, said she had been subject to similar hormonally themed attacks throughout her career.

“As I made my way up in the business world, a male-dominated business world, I’ve had lots of men imply that I was unfit for decision-making because maybe I was having my period,” she said on CNN’s “State of the Union.”

Mrs. Fiorina praised private sector companies such as Netflix Inc. that have offered workers paid maternity leave, though she reiterated her opposition to “the federal government mandating paid maternity leave to every company out there.”

She said Netflix NFLX -2.37% was “doing the right thing because they know it helps them attract the right talent.”

Ohio Gov. John Kasich said Sunday that he went out of his way to recruit women to his campaign and administration because, in his experience, “whenever women touch anything, they clearly make it better than we do, as guys.”

“I have got strong women in my family. I have got strong women in my administration. And I have strong women in my campaign. In fact, my campaign manager is a woman,” he said.

Mr. Trump, a real estate magnate, also sought to position himself as a champion for women in the workplace, even as he refused to apologize for attacking the appearance of women who had criticized him.

He praised his female employees as “amazing executives,” “phenomenal,” and “killers.”

“I’ve always had a great relationship to the women,” he said on ABC’s “This Week.”  “I was one of the first people in the construction industry to put women in charge of major construction projects.”

Meanwhile, GOP presidential contenders former Arkansas Gov.  Michael Huckabee and Florida Sen. Marco Rubio both affirmed their strict anti-abortion positions.

“I don’t think it’s legal,” Mr. Huckabee said of abortion on ABC’s show. “I think it violates the Fifth and Fourteenth Amendment rights of an unborn person.”

Mr. Rubio said there was “no doubt that a woman has a right to her own body,” but that right needed to be balanced with “the right of a human being to live.” He illustrated what he called a “difficult question” with a hypothetical scared 15-year-old girl who needed an abortion, concluding that he would “err on the side of life,” suggesting he would have the girl carry the pregnancy to term.

Sabrina Schaeffer, executive director of the conservative research group Independent Women’s Forum, said she hoped Sunday’s discussion of women in the workplace heralded the beginning of a broader conversation among conservatives on how to reach women voters on economic issues such as pay equity and flexible working arrangements.

“The reality is, I don’t think that many conservatives are paying attention to how much Democrats recognize the importance of women voters [who] have really shifted their focus to workplace issues,” she said in an interview. “I think that’s a real opportunity for Republicans not to shy away from talking about those issues but to take them head on.”

Democratic National Committee spokeswoman Holly Shulman dismissed the Republican candidates’ remarks.

“Rubio tripled down on his anti-choice policies – without exceptions. Kasich listed the binders of women he knows, but his policies have shut down women’s health clinics,” she said in a statement.  “It’s not just Republican rhetoric that is offensive to women, it’s their policies as well. And today’s Sunday show appearances only made things worse.”