Prime Minister Boris Johnson of Britain was battling for his political survival on Tuesday when two senior ministers in his Conservative government unexpectedly resigned their cabinet posts in what appeared to be a coordinated move against their leader.

Rishi Sunak, the chancellor of the Exchequer, and Sajid Javid, the health secretary, both quit over the latest scandal to raise questions about Mr. Johnson’s judgment and honesty. They announced their decisions shortly after Mr. Johnson apologized for appointing a minister, Chris Pincher, who last week quit his job over accusations of inappropriate behavior.

The resignations thrust Mr. Johnson into the most perilous position of his three-year tenure as prime minister, after a series of crises that forced him to survive a vote of no confidence last month.

Unless the party’s rules are changed, he cannot face another contest for a year. That means that resignations from the cabinet could be the only effective method of putting pressure on Mr. Johnson to resign.

Mr. Johnson’s fate may rest on whether other members of his cabinet decide to stand by him, or join a rebellion that could make Mr. Johnson’s position untenable. Though a number of senior ministers are known to be loyal to the prime minister, there was doubt about some others whose statements and actions will be watched closely.

If Mr. Johnson can avert further departures, that would improve his chances of surviving. But more resignations would heap massive pressure on the prime minister and would suggest that his government was collapsing.

Even if the remainder of the cabinet stays loyal, Mr. Johnson faces an uphill battle to restore his battered authority. Even before the resignations on Tuesday evening, Conservative lawmakers were speculating that there could be a swift change to the party’s rulebook that would allow a fresh no-confidence vote before the summer recess.

So far Mr. Johnson has refused calls for his resignation and, in a B.B.C. interview on Tuesday, he appeared to believe that an apology would bring his restive lawmakers to heel. Mr. Johnson admitted it was “a mistake” to make Mr. Pincher deputy chief whip given previous complaints about his conduct. “With hindsight it is the wrong thing to do and I apologize to everyone who has been badly affected by it,” he added.

But his critics believe they have heard similar assurances previously.

“The public rightly expect government to be conducted properly, competently and seriously,” said Mr. Sunak in a statement. “I recognize this may be my last ministerial job, but I believe these standards are worth fighting for and that is why I am resigning.”

Mr. Pincher resigned his position as deputy chief whip last week after admitting to having been drunk at a private members club in London where, according to British media reports, he groped two men. He was suspended from the Conservative Party while the allegations are being investigated but he has not resigned as a member of Parliament.

On Tuesday, Downing Street admitted that Mr. Johnson had been told about previous allegations against Mr. Pincher in 2019 — something that Mr. Johnson’s office had initially denied.

The outcry over the circumstances of the appointment of Mr. Pincher — and Downing Street’s account of it — is just the latest in a series of scandals surrounding Mr. Johnson. Earlier this year, he was fined by the police for breaking his own lockdown rules in Downing Street, in which members of his staff were found to have held a number of alcohol-fueled parties.

In his resignation statement, Mr. Javid said that he could “no longer, in good conscience, continue serving in this government.” The public, he added, expects “government to be conducted properly, competently and seriously.”