For anyone still wondering whether President Trump would tone down his personal attacks in response to perceived slights as he settled in the White House, his tweets Thursday mocking a female cable news host seem to have provided an answer.

The longer-term danger of Trump’s tweets about Mika Brzezinski and her MSNBC “Morning Joe” co-host Joe Scarborough is that he may be codifying personal attacks as the new normal because they come from the White House.

In a pair of tweets early Thursday, Trump wrote: “I heard poorly rated @Morning Joe speaks badly of me (don’t watch anymore). Then how come low I.Q. Crazy Mika, along with Psycho Joe, came ... to Mar-a-Lago 3 nights in a row around New Year’s Eve, and insisted on joining me. She was bleeding badly from a face-lift. I said no!”

What apparently set Trump off was Brzezinski’s references to a Washington Post story about a fake Time magazine cover featuring Trump that hangs in several of his golf resorts. “Nothing makes a man feel better than making a fake cover of a magazine about himself, lying every day and destroying the country,” Brzezinski said on the show Thursday morning.

Some may dismiss this as a spat between an unpredictable president and highly paid cable hosts who used to be on friendlier terms. The show gave Trump a lot of positive airtime during the primary, and he even offered to officiate the newly engaged couple’s wedding in the White House, according to Scarborough.

But Thursday’s tweet may reveal something deeper about Trump.

“People were wondering if he would become less sexist and misogynistic when he entered the White House. But the reality is you just don’t change when you get into office,” said Kelly Dittmar, a Rutgers University political science professor and scholar at its Center for American Women and Politics who studies gender issues. “This is a pattern of behavior for him. He gets particularly irked when women challenge him.”

Like when then-Fox News host Megyn Kelly challenged him during a GOP campaign debate for referring to women as “fat pigs,” “dogs,” “slobs” and “disgusting animals.” During an interview afterward, Trump said, “You could see there was blood coming out of her eyes. Blood coming out of her wherever.”

Men have used this sort of “gendered” attack — calling a woman crazy or referring to her personal appearance — “against women who are critical of them for hundreds of years,” Dittmar said.

Thursday’s comment was “significant because the president of the United States is a symbol of the values we hold as a nation,” she said. Hearing Trump use that language “is one way of showing how we talk about women, how we value women.”

For her part, Brzezinski responded by diving into the rhetorical gutter with the president, tweeting a photo of a Cheerios ad that read, “made for little hands.” It was a reference to the size of Trump’s hands that Republican Sen. Marco Rubio made during a campaign debate last year.

And while there was plenty of outrage from women and men alike Thursday, none of the women closest to Trump objected publicly to his harshest personal attack on a woman since taking office.

Not spokeswoman Sarah Huckabee Sanders, who said the president was “pushing back against people who have attacked him day after day after day. Where is the outrage on that?”

“The American people elected a fighter; they didn’t elect somebody to sit back and do nothing,” Sanders told reporters.

During a campaign speech in November, Trump’s wife, Melania, promised that as first lady she would work to stop cyberbullying. But on Thursday, her spokeswoman defended the president’s comments about Brzezinski’s looks and intelligence, saying, “When her husband gets attacked, he will punch back 10 times harder.”

And there were no objections from Trump’s daughter and adviser Ivanka Trump, who represented the White House at an April forum in Germany for women’s economic empowerment and has advocated for female entrepreneurs.

Two women who did speak out against Trump on Thursday could have more of an impact, because they’re key to passing the Republican health care legislation that’s stalled in the Senate.

Sen. Susan Collins, R-Maine, who is not in favor of the current measure, tweeted, “This has to stop — we all have a job — 3 branches of gov’t and media. We don’t have to get along, but we must show respect and civility.”

Sen. Lisa Murkowski, R-Alaska, who is undecided on the measure, tweeted, “@POTUS, do you want to be remembered for your tweets or your accomplishments?”

But it’s unlikely that either senator will base her health care vote on a Trump tweet, said Brendan Nyhan, a Dartmouth College political science professor and co-author of “All the President’s Spin: George W. Bush, the Media and the Truth.”

“If they were going to turn against Trump based on his personal behavior, they would have done so a long time ago,” Nyhan said.

It’s also unlikely that this latest Twitter outburst will affect Trump’s standing with voters. Through a litany of crass attacks on opponents over the past two years, Trump has managed not only to win the presidency but whip up enough enthusiasm that a chunk of the public sees them as the new normal.

Little seems to dent the 40 percent of voters who approve of the president or the 54 percent who disapprove, according to the latest amalgamation of top polls by RealClearPolitics.com. Voters care less about what Trump tweets, Nyhan said, and “more about the economy, the state of our foreign policy, the federal health care bill.”

“At this point everyone has made up his mind,” he said. “It’s hard to surprise people with what he says anymore. People may find it outrageous — people who don’t like him, that is.

“We’ve never seen a president go so far,” Nyhan said. “When there are high-profile figures or institutions that are critical of the president, whether it is a federal judge or a (former FBI director) James Comey, he lashes out at them and attacks them.”

And then there is this minor note: Trump’s tweet was also factually inaccurate. “Morning Joe” just experienced its highest-ever ratings in the year’s second quarter, with its nearly 1 million morning viewers, and beating its rivals on CNN, but finishing behind Fox News.

Joe Garofoli is The San Francisco Chronicle’s senior political writer. Email: jgarofoli@sfchronicle.com Twitter: @joegarofoli