Improbably but relentlessly, Donald Trump is marching toward the Republican Party’s presidential nomination, a tepid second-place finish in Iowa notwithstanding. Last week, Trump showed it’s possible to win a debate by not showing up.

But Trump has an Achilles’ heel when it comes to winning a general election: He doesn’t do well at attracting women voters. Actually, that’s an understatement: He repels many of them.

In recent polls of Republican voters in Iowa, the boastful mogul was the first choice of 37 percent of men (when results were averaged) and 31 percent of women. Because there are more male Republican voters than women — 57 percent in the 2012 Iowa caucuses, for example — that gender gap, while sizable, is not fatal.

In the electorate as a whole, however, there are more female voters than male — 53 percent in 2012 — and there, Trump’s gender gap is more pronounced.

When CNN asked voters whether they had a favorable or unfavorable impression of Trump, men were almost evenly split: 44 percent favorable, 47 percent unfavorable. But a striking 64 percent of women said they did not like him.

Why is Trump struggling to win women’s hearts?

“I will be the best thing that ever happened to women,” he said last year. “I cherish women.” That didn’t do the trick.

Maybe it’s the endless locker-room braggadocio. Maybe it’s the insults he sprays at opponents, male and female alike. Or maybe it’s just the persistent appearance of misogyny.

This is, after all, a man who called Fox News anchor Megyn

Kelly a “bimbo” and said she had “blood coming out of her wherever,” who sent a note to a New York Times columnist telling her she had “the face of a dog,” and whose professional assessment of Carly Fiorina was: “Look at that face! Would anyone vote for that?”

Katie Packer, a former Mitt Romney campaign aide who runs a political action committee opposing Trump, argues that the front-runner’s woman problem stems from more than old-fashioned sexism.

“One of the Republican Party’s problems is that it is seen by a lot of voters — younger voters, female voters, minority voters — as a bunch of grouchy old rich white guys,” she said. “Trump is old, rich and grouchy — and he’s a dude.”

Another issue — although it’s one that hasn’t been mentioned much so far — is Trump’s dramatic marital history: Two of his three splashy marriages ended in equally splashy divorces.

The key to a successful relationship, the candidate has said, is a good prenup.

His first marriage, to Czech model Ivana Zelnickova, collapsed after she discovered that he was having an affair with a younger woman, Marla Maples. In a scene worthy of reality TV, Ivana confronted Marla on a ski slope in Aspen before dozens of witnesses.

Maples later paid Trump a compliment that showed up on the front page of the New York Post: “Best Sex I Ever Had.” Not clear whether that’s a plus or minus in a presidential campaign. She became Trump’s second wife two months after the birth of their daughter, Tiffany.

Now Trump is on spouse number three, Slovenian ex-model Melania Knauss, whom he married in 2005 when he was 58 and she was 34.

Is Trump’s personal history a legitimate target in a presidential campaign? Of course — just like any other candidate’s.

That’s a principle Trump himself acknowledged last month when he promised to attack Bill Clinton’s conduct toward women.

“If Hillary Clinton is going to play the woman card, it’s all fair game,” he said.

Be careful, Donald. That’s an argument you might not win.

When the Fox News poll asked voters whether they thought Trump or Bill Clinton was “more respectful of women,” Clinton won, 50 percent to 37 percent. Women were even more emphatic; 55 percent said they thought Clinton was more respectful. But there was a partisan split; Republican women preferred Trump — more evidence that Trump could win the GOP nomination but face an uphill struggle in the general election.

With all those complications, why is Trump doing so well?

“He’s figured out the mood of the voters,” said David Winston, a Republican pollster who is not affiliated with a campaign. “People want to hear how a candidate plans to change everything. They are looking for someone who will rock the boat. Trump is doing that more clearly than the others.”

That could be just enough to allow Trump to win the Republican nomination. But thanks to the 19th Amendment, it doesn’t look as if it’s enough to win the presidency.

Doyle McManus is a columnist with the Los Angeles Times.