WASHINGTON — When Anthony Scaramucci, the new White House communications director, went on television on Thursday morning to compare himself and his adversary, Reince Priebus, the chief of staff, to Cain and Abel, it seemed to encapsulate the fratricidal nature of an administration riven by biblical rivalries. Cain, after all, killed Abel as they vied for God’s favor.

As it turned out, that was the cleaned-up version. In a vulgarity-laced telephone call with a New Yorker writer reported on the magazine’s website on Thursday evening, Mr. Scaramucci railed against Mr. Priebus and Stephen K. Bannon, the president’s chief strategist, both of whom opposed his hiring last week. He even vowed to get the chief of staff fired. “Reince Priebus — if you want to leak something — he’ll be asked to resign very shortly,” Mr. Scaramucci said.

Whether Mr. Scaramucci will turn out to be Cain or Abel, it was clear that his appointment has added another layer of drama and dissent to a White House suffused in it — and revived the perpetual questions about Mr. Priebus’s fate. Sean Spicer, the White House press secretary and an ally of Mr. Priebus, resigned in protest when Mr. Scaramucci was hired last week because, he predicted, it would only add more chaos to the team. On that, at least, he seems to have been proved right.

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But President Trump not only tolerates feuds within his team, he fuels them, playing one courtier off another and leaving them all unsteady. He chooses favorites and casts others aside, but even those decisions seem subject to change at any moody moment. And by several accounts, he personally encouraged Mr. Scaramucci’s jihad against Mr. Priebus, once again subjecting his chief of staff to a ritualistic public lashing even as he considered pushing him out.

Left to explain all this was Sarah Huckabee Sanders, the new White House press secretary. “This is a White House that has lots of different perspectives because the president hires the very best people,” she said gamely, before the New Yorker article posted, asserting that a “healthy competition” benefits Mr. Trump. “With that competition, you usually get the best results. The president likes that kind of competition and encourages it.”

That kind of competition has exhausted even some of Mr. Trump’s most loyal defenders. But Mr. Trump has openly told people that he has lost faith in Mr. Priebus. He has said he wants “a general” as chief of staff, and has focused on John F. Kelly, the retired four-star Marine now serving as homeland security secretary. Many of his advisers, however, consider that a bad idea.

Mr. Scaramucci made clear in his conversation with The New Yorker’s Ryan Lizza that he is trying to push Mr. Priebus out. “Reince is a fucking paranoid schizophrenic, a paranoiac,” he said. Mr. Scaramucci complained that Mr. Priebus had prevented him from getting a job in the White House until now, saying he “blocked Scaramucci for six months.”

In the same telephone call, Mr. Scaramucci disparaged Mr. Bannon. “I’m not Steve Bannon. I’m not trying to suck my own cock,” he said. “I’m not trying to build my own brand” on the president’s coattails.

“I’m here to serve the country,” he added.

Mr. Priebus finds himself isolated inside the White House. He has lost the support of Mr. Trump’s family, and other senior aides have long bristled at his demeanor or suspected he was trying to undermine them. Allies like Mr. Spicer are gone or leaving. And some complain that Mr. Priebus used the White House communications office as his own personal fiefdom.

Lately Mr. Trump has resumed subjecting him to frequent indignities in front of White House staff. According to one aide, the president, who had ceased for a time, has regularly mentioned how Mr. Priebus suggested that Mr. Trump consider dropping out of the presidential race last October after a tape of him boasting about grabbing women by the genitals emerged. “Do you remember when Reince did that?” the president has asked associates. The issue has always been a sore spot between the two men.

Mr. Priebus endured the hazing in silence, as he generally has, and the White House did nothing to defend him against Mr. Scaramucci’s tirade. Mr. Scaramucci released a statement after the New Yorker piece was published that fell well short of an apology.

“I sometimes use colorful language,” he said on Twitter. “I will refrain in this arena but not give up the passionate fight for @RealDonaldTrump’s agenda.”

Ms. Sanders said mildly that Mr. Scaramucci was simply expressing strong feelings, and that his statement made clear that “he’s a passionate guy and sometimes he lets that passion get the better of him.” She added, “I don’t think he’ll do it again.”

Mr. Priebus’s plight was good news for another member of the Trump team. For the first time in a week, it was not Attorney General Jeff Sessions’s turn to be the presidential punching bag.

During a visit to El Salvador, Mr. Sessions acknowledged to the Associated Press that “it hasn’t been my best week” in his “relationship with the president.” Speaking to Fox News, he added, “It’s kind of hurtful, but the president of the United States is a strong leader. He is determined to move this country in the direction that he believes it needs to go to make it great again.”

So many figures inside Mr. Trump’s orbit have been declared on their way out that it takes a scorecard to keep track. Aside from Mr. Priebus and Mr. Sessions, many wonder about the future of Lt. Gen. H. R. McMaster, the national security adviser whose Afghanistan war plan was rejected by the president last week. Secretary of State Rex W. Tillerson disappeared for a few days off, stoking speculation that he may leave. (“Rexit,” it was dubbed on Twitter.) And the president, who has already fired one F.B.I. director, this week called for the acting head of the bureau to be dismissed too.

The clash between Mr. Scaramucci and Mr. Priebus offers a case study in how the Trump White House operates, a conflict divorced from facts, untethered from the basics of how government works, enabled by the lack of any organizational structure and driven by ambition, fear, animosity and envy.

The genesis was a dinner hosted Wednesday night by Mr. Trump at the White House that included Mr. Scaramucci, Sean Hannity and Kimberly Guilfoyle, the Fox News hosts, and Bill Shine, a former Fox executive.

Ms. Guilfoyle told the president that Mr. Priebus was a problem and a leaker, someone who was not serving his agenda, according to a person briefed on the conversation.

Mr. Scaramucci grew angry afterward that Mr. Lizza had learned that the dinner was taking place and that Politico had obtained his government financial disclosure form. At that point, he called Mr. Lizza, demanding to know his source, whom the reporter refused to divulge.

“O.K., I’m going to fire every one of them, and then you haven’t protected anybody, so the entire place will be fired over the next two weeks,” Mr. Scaramucci replied.

After hanging up, Mr. Scaramucci posted a message on Twitter asserting that the “leak” of his disclosure form was a “felony” and that he would seek an F.B.I. investigation. He included Mr. Priebus’s Twitter handle, a move that was interpreted as blaming the chief of staff.

But it was no leak. The disclosure form is supposed to be made public under federal law and all Politico did was ask for it under normal procedures. Mr. Scaramucci deleted the tweet. But on Thursday morning, he called into CNN with Mr. Trump’s encouragement, and threw down the gauntlet with Mr. Priebus on live television.

“We have had odds. We have had differences,” Mr. Scaramucci said on CNN. “When I said we were brothers from the podium, that’s because we’re rough on each other. Some brothers are like Cain and Abel. Other brothers can fight with each other and get along. I don’t know if this is reparable or not. That will be up to the president.”

Some of Mr. Trump’s supporters said Mr. Scaramucci was causing more harm than good.

“I would say right now that he’s being more pugnacious than effective,” Newt Gingrich, the former House speaker, told the radio host Laura Ingraham. “I think he ought to slow down a little bit and learn what he’s doing.”