The N.H.L. fined the Rangers $250,000 on Thursday for the team’s unusually personal comments about a league player safety official this week.

The Rangers had publicly rebuked the head of the N.H.L.’s department of player safety, George Parros, on Tuesday for his failure to suspend Washington Capitals forward Tom Wilson after he punched Rangers forwards Artemi Panarin and Pavel Buchnevich during a scrum in Monday’s game at Madison Square Garden.

The league on Tuesday fined Wilson $5,000, the maximum allowable amount, and the Rangers later released a scathing statement criticizing the N.H.L. for not issuing harsher discipline and saying Parros was “unfit” to continue in his role.

On Thursday, a day after a Rangers-Capitals rematch in New York was marred by brawls, the league’s commissioner, Gary Bettman, weighed in on the dispute.

“Public comments of the nature issued by the Rangers that were personal in nature and demeaning of a league executive will not be tolerated,” Bettman said. “While we don’t expect our clubs to agree with every decision rendered by the department of player safety, the extent to which the Rangers expressed their disagreement was unacceptable.”

The fine came after a day of upheaval for the Rangers on Wednesday, which started with the ouster of two top executives and ended with a game between two teams that had a score to settle.

Hours after the firing of John Davidson, the team president, and Jeff Gorton, the general manager, the Rangers began their game against the Capitals with a choreographed show of vengeance that led to three simultaneous fights after the opening face-off.

The teams combined for 100 penalty minutes in the first period. At one point there were more Capitals in the penalty box than on the ice.

Furious at the league, at Wilson and at being out of playoff contention, the Rangers appeared ready to fight. As soon as the puck dropped to start the game, Rangers center Kevin Rooney fought with Nic Dowd; winger Phil Di Giuseppe battled Washington’s Garnet Hathaway; and Colin Blackwell traded punches with Capitals forward Carl Hagelin.

It took only 50 seconds for Rangers defenseman Brendan Smith to confront Wilson, shoving him into the boards at the Rangers’ bench before the two began trading punches. There were six fights in the first five minutes.

“I had no beef with anybody else on their team,” Smith said of his focus on fighting Wilson early. “I thought it should have been handled before this game, and it wasn’t. I felt it had to be on my shoulders, and I took it.”

Wilson was loudly booed by the several thousand fans in the Garden every time he was on the ice, which wasn’t for long. He played 2 minutes 36 seconds and accumulated 15 penalty minutes before leaving the game with an upper-body injury.

The Rangers lost their fourth game in a row, 4-2, and have two games remaining. They were eliminated from playoff contention on Monday, when they lost, 6-3, to the Capitals, who are headed to the postseason.

Off the ice, the Rangers’ management shake-up has begun. Chris Drury, 44, the associate general manager, will become the team’s new president and general manager. Drury, the N.H.L. rookie of the year in 1999, spent the last four seasons of his playing career with the Rangers and joined the team’s front office in 2015. Glen Sather, who has been a senior adviser with the team since 2019, will work with Drury during the transition period.

The Rangers have not made the traditional N.H.L. playoffs since the 2016-17 season, though they participated in last year’s postseason format, which was expanded because the coronavirus pandemic had interrupted the season.

“We want to thank J.D. and Jeff for their contributions to the organization,” James L. Dolan, the team owner, said in a statement announcing the firings of Davidson and Gorton. “They are both great hockey professionals who worked hard for the Rangers, however, in order for the team to succeed in the manner our fans deserve, there needs to be a change in leadership.”

Davidson said at his introductory news conference two years ago that he was eager to work with Gorton and Coach David Quinn to develop a culture of accountability with the Rangers. Quinn, the former men’s hockey coach at Boston University, is finishing his third season behind the Rangers’ bench.

Gorton, 52, and Davidson, 68, were overseeing a rebuild that began in earnest with a letter sent to the fan base in February 2018. Since then, the Rangers have made six first-round draft picks, including the heralded forwards Kaapo Kakko at No. 2 in 2019 and Alexis Lafreniere at No. 1 in 2020.

“There are no quick fixes,’’ Davidson said when he was hired. “It’s hard work, it’s patience and it’s having the proper resolve.”