By Brian Stelter, CNN

(CNN) -- The White House Correspondents' Association is the newest front in President Donald Trump's long-running war with the media.

The President continued to slam the organization and its annual dinner on Monday morning, tweeting that the event is "DEAD as we know it. This was a total disaster and an embarrassment to our great Country and all that it stands for. FAKE NEWS is alive and well and beautifully represented on Saturday night!"

On Sunday, he called comedian Michelle Wolf -- who ripped into Trump and his aides on national television -- "filthy" and suggested that the organization "put Dinner to rest, or start over!"

That's unlikely to happen. But the association did issue a rare statement on Sunday night expressing regret about Wolf's performance.

The association's president, Margaret Talev, did not apologize, as some Trump allies and Washington journalists wanted, but she said the roast wasn't in keeping with the group's mission.

"Last night's program was meant to offer a unifying message about our common commitment to a vigorous and free press while honoring civility, great reporting and scholarship winners, not to divide people," Talev said. "Unfortunately, the entertainer's monologue was not in the spirit of that mission."

But others are defending Wolf and saying she spoke truth to power. The debate is a microcosm of ongoing arguments over the proper tone of Trump White House criticism.

Wolf's raunchy routine, which included jabs at White House press secretary Sarah Sanders, counselor to the president Kellyanne Conway and the news media itself, was met with a mixture of laughter and uncomfortable silence from the audience of journalists and guests.

Come Sunday morning, there was lots of second-guessing. One side bemoaned liberal intolerance. The other side assailed conservative victimhood. Outrage over Wolf's jokes about Sanders was met with criticism of Sanders' performance at the briefing room podium.

Fox News' Ed Henry, NBC News' Andrea Mitchell and other prominent correspondents called on the association to apologize for the most controversial portions of Wolf's act.

But Talev stopped short of apologizing in an appearance on CNN's "Reliable Sources."

She acknowledged that "the comedian reflects on the press corps, but I don't think that the comedian speaks for the press corps."

Talev, a Bloomberg News correspondent who doubles as a CNN analyst, said she had one regret: "To some extent, those 15 minutes" of comedy "are now defining four hours of what was a really wonderful, unifying night. And I don't want the cause of unity to be undercut."

Talev also pointed out that comedians routinely aim to shock and surprise. "Comedy is meant to provoke thought and debate. And it certainly has done that," she said.

Talev's Sunday night email to members indicated that changes could be afoot. She said she and SiriusXM's Olivier Knox, the association's next president, "are committed to hearing from members on your views on the format of the dinner going forward."

Trump's tweet came about half an hour later.

"The White House Correspondents' Dinner was a failure last year, but this year was an embarrassment to everyone associated with it," he wrote.

Trump chose to skip both dinners, breaking with decades of presidential precedent, and instead hold a political rally.

The dinner's organizers describe it as a celebration of the First Amendment.

Trump's Sunday night tweet was also notable because he didn't call out Wolf by name -- he only said "the filthy 'comedian' totally bombed."

Trump added, "she couldn't even deliver her lines -- much like the Seth Meyers weak performance."

Meyers was the comedian at the association's 2011 dinner. That was the time Trump attended the dinner as a guest of The Washington Post. Both Meyers and President Barack Obama harshly criticized Trump in their remarks. The dinner has since been cited as a motivating factor for Trump's 2016 run for office.