After Senator Marco Rubio of Florida insisted at the Republican presidential debate that rape and incest victims should carry pregnancies to term, aides to Hillary Rodham Clinton could barely contain their delight at his unyielding stance, rushing to tell reporters at her headquarters that those remarks would hurt Mr. Rubio with female voters.

When Donald J. Trump chose on Friday to stand by his slights against women during the debate, saying the Fox News journalist Megyn Kelly “behaved very badly” as a moderator — and then promoting a Twitter message calling her a “bimbo” — feminists were not the only ones outraged: the chairwoman of the New Hampshire Republican Party accused Mr. Trump of chauvinism.

And in response to multiple male candidates saying they would shut down the federal government over financing for Planned Parenthood, the Democratic National Committee emailed talking points to allies within an hour saying that among the losers at the debate were “American women, who were attacked at every turn.”

Republican Party leaders, whose presidential nominees have not won a majority of female voters since 1988, are setting their sights on making electoral gains among women in the 2016 presidential race and trying to close the gender gap in swing states like Florida and Colorado. But the remarks and tone about women at Thursday’s debate — and the sight of 10 male candidates owning the stage — may have only damaged the party’s standing among female voters in the 2016 general election, according to pollsters and some Republican leaders.