Senator Ted Cruz may have urged Donald J. Trump to leave his wife, Heidi, “the hell alone,” but there is one group that is quietly hoping Mr. Trump’s attacks on his rival’s spouse and other women will continue indefinitely: Democrats.

As Hillary Clinton turns her attention to a general election campaign, Mr. Trump’s nasty skirmish with Mr. Cruz, including his warning to “spill the beans” about Mrs. Cruz, without offering specifics, and his reposting of a message that mocked her looks, have played into a crucial Democratic strategy to defeat Mr. Trump in November: to portray him as an unabashed sexist.

Mrs. Clinton’s allies hope to sway suburban and independent women, who will play an outsize role in deciding the fall election, to support her candidacy by pushing this theme. These Democrats say the matchup would be historic: one pitting the first female nominee of a major political party against a rival who has repeatedly dismissed and disparaged women and their looks.

“I want Donald Trump to talk every single day for the rest of this election,” said Representative Debbie Wasserman Schultz of Florida, the chairwoman of the Democratic National Committee. “He just needs to keep spewing what he has been spewing.”