At first blush, the instantly viral hot-mic incident at the White House on Monday, in which President Biden called a Fox News reporter “a stupid son of a bitch” on live television, presented a grim object lesson about today’s political-media complex: When vulgarity happens, everybody wins.

Liberal Biden fans cheered (“literal lol,” per the MSNBC host Chris Hayes). Fox News pundits pounced on a new grievance (“What a nasty old man,” said the host Tucker Carlson). Newspapers got clicks (The New York Times’s version of the story was among the most-read on the paper’s website) and cable news found fresh chum for its 24/7 news processing plant.

But the spectacle of Mr. Biden lobbing an unintentionally amplified expletive at Peter Doocy, the fresh-faced Fox News reporter and a regular foil, also turned out to be one of the most unlikely feel-good moments of his time in office.

OK, hear me out.

Yes, it’s never a good thing for democratic discourse when the most powerful man in the country insults a reporter for doing his job. (See Trump, Donald J., many examples from 2015 to present.) Yes, children probably should not overhear the president mutter such language live on television. Yes, coarse insults and ad hominem attacks are helping to fuel what sometimes feels like a hopeless partisan divide.

But what happened afterward was weirdly heartening.

Mr. Doocy, 34, beamed into Fox News from the White House lawn and … laughed about it. “Nobody is fact-checking yet and said it’s not true,” he said with a self-deprecating grin, defying the entreaties of network colleagues like Sean Hannity to indulge in a Biden-bashing victory lap.

And the president did something that his predecessor, Mr. Trump, never seemed to contemplate after one of his tirades against the press: He called up Mr. Doocy and cleared the air.

“He said, ‘It’s nothing personal, pal,’” Mr. Doocy said of the call from Mr. Biden, which arrived within an hour of the president’s uncouth remark. “We were talking about just, kind of, moving forward.”

Bob Schieffer, the veteran CBS News anchor, surveyed the exchange from the vantage point of having covered 10 presidents.

“It was something we don’t see that often: a civil exchange,” Mr. Schieffer said in an interview. “Both of them came off looking the better for it.”

Part of Mr. Doocy’s calculus, surely, was remaining cordial with the White House, which he covers every day and whose staff members are important gatekeepers for his ability to do his job. And aides to Mr. Biden have never seen much advantage to being drawn into active warfare with Fox News, preferring to keep the vitriol one-way and avoid alienating the independent and persuadable members of the network’s vast audience.

Mr. Biden’s allies have also expressed concern that his relatively few formal interviews with the news media have given critics too much room to define him. An unfiltered moment, inadvertent or not, can remind voters of what they liked about him as a candidate — a politician at his best when he is open with his emotions, even the frustrations of a tough day at the office.

“They were both adults,” said April Ryan, the White House correspondent for TheGrio, who weathered a number of nasty remarks from Mr. Trump, none of which required a hot mic to be heard. “Biden did the right thing and apologized, and Peter was gracious and accepted that. And that’s it. The apology and then the acceptance and the moving on was the best scenario of an ugly situation.”

Plus, even the harshest partisans got a kick out of it.

“Don’t tell my fellow right-wingers, but I thought what Biden said was funny,” Ann Coulter, the arch and arch-conservative pundit, wrote in an email. Although Ms. Coulter could not resist taking a shot at Mr. Biden’s mental fitness, she also conceded, approvingly: “He’s still got sarcasm!”

While the exchange echoed past presidents’ venting about the press, some around Washington pointed out that its resolution showed just how different things are under Mr. Biden than his predecessor.

Mr. Trump not only viciously insulted reporters (“disgrace”; “loser” and “enemy of the people”), he also incited public anger against the press. His supporters sent frightening threats to journalists and in at least one case mailed a pipe bomb to CNN. Freedom of the press groups have credited his behavior with eroding the rights and privileges of journalists in autocratic countries around the world.

Jim Acosta, a CNN correspondent who was often on Mr. Trump’s radar, at times seemed to relish his role as an antagonist. His 2020 memoir, called “The Enemy of the People,” was described in marketing materials as “an explosive, firsthand account of the dangers he faces reporting on the current White House.”

It is possible that Mr. Doocy could have his own book soon enough. (A riff on Mr. Biden’s comment might make for a blunt, if eye-catching, title). But in a brief interview on Thursday, he signaled a more conciliatory approach.

“It’s important — whatever people think of me and Biden — to see that we can have a quick phone call and resolve things, that he and I could just chat,” Mr. Doocy said. He said he has received an enormous response to the episode, including calls from long-lost elementary school classmates.

Mr. Schieffer said Mr. Biden’s gaffe reminded him of his days covering George McGovern’s 1972 presidential campaign for CBS. The Democratic candidate was greeting voters in Michigan when a local heckler began taunting him (accurately, it turned out) about his impending loss to Richard Nixon.

“McGovern called him over and, loud enough that we could hear it, said, ‘Hey, kiss my ass,’” Mr. Schieffer recalled.

It would take about 15 seconds in today’s world for such an unguarded remark to start blanketing Twitter and CNN panel discussions. Instead, the political correspondents “spent all afternoon, the rest of the day, trying to figure out if we could use the word ‘ass’ in our reports.”

(The Times, in its own account of the McGovern incident, which took place in Battle Creek, Mich., in November 1972, referred to the candidate’s remark as “a vulgarism.” And like Mr. Biden, McGovern later offered regrets, telling a journalist, “We need a little more courtesy in this country, and I am going to practice a little more myself.”)

Mr. Schieffer, who still occasionally contributes to CBS but is spending much of his time painting, said that in the end, Mr. Biden’s gaffe was “not that big a deal.”

“In the course of American history,” he said, “this is not going to get a lot of second-day play, if you know what I mean.”