President Trump’s Twitter account again stole the spotlight Sunday, drawing fire from both sides of the aisle as Republicans stepped up their rebukes and claimed the commander in chief is using social media as a key part of a sinister strategy to “weaponize distrust” in his No. 1 enemy: the media.

While Mr. Trump has been waging war against the press since the early days of his campaign, it reached new heights over the past week, culminating in a Sunday morning presidential tweet of a doctored video of him attacking WWE chief Vince McMahon, with the CNN logo superimposed over Mr. McMahon’s face.

The footage was taken from WrestleMania 23 in 2007, and the video led critics, including CNN, to claim Mr. Trump was overtly encouraging violence against reporters.

Mr. Trump took personal shots at MSNBC host Mika Brzezinski via Twitter last week and castigated the press during a speech this weekend.

Even some Republicans say the president’s testy relationship with the Fourth Estate has gone too far. Sen. Ben Sasse, Nebraska Republican and frequent critic of Mr. Trump, said Sunday that the president is using his social media prowess to lead America down a dangerous path.

“There’s an important distinction to draw between bad stories or crappy coverage, and the right citizens have to argue about that and complain about that — and trying to weaponize distrust. The First Amendment is the beating heart of the American experiment, and you don’t get to separate the freedoms that are in there,” Mr. Sasse said.

The freshman Republican suggested that many Americans, including Mr. Trump, are encouraging or perpetuating a social compact in which citizens listen only to those who agree with their political views and that the bedrock idea of absolute, undisputable facts on which everyone can agree is going by the wayside.

“The reality is journalism is really going to change a lot more in the digital era, and we have a risk of getting to a place where we don’t have shared public facts. A republic will not work if we don’t have shared facts,” he said. “It is going to be possible in the next three or five or 10 years for people to surround themselves only with echo chambers and silos and people who believe only what they already believe. That’s a recipe for a new kind of tribalism, and America won’t work if we do that.”

Ohio Gov. John Kasich, a Republican and another outspoken critic of the president, said Mr. Trump’s conduct “is not acceptable.”

“It’s unfortunate, and people are now begging the president not to do this and, you know, he ought to stop doing it and we’ll have to see what happens,” he said on ABC’s “This Week” program. “It’s one of the few things that I think brought Republicans and Democrats together. They spend so much time fighting and then they’re all aghast, you know, and so it’s just not the way we ought to be. The coarseness is not acceptable.”

Mr. Kasich’s and Mr. Sasse’s comments Sunday were some of the harshest offered by members of the president’s own party, though other Republicans also expressed disgust over the past week.

Sens. Lisa Murkowski of Alaska, Susan M. Collins of Maine, Lindsey Graham of South Carolina, James Lankford of Oklahoma and others strongly condemned the attack against Ms. Brzezinski, in which Mr. Trump claimed that the “Morning Joe” co-host was “bleeding badly from a face-lift” at the president’s Florida resort on New Year’s Eve.

Ms. Brzezinski vehemently denied that claim and, along with co-host Joe Scarborough, continued blasting Mr. Trump and questioning his mental health and fitness for office. Ironically, those were the very criticisms that Trump supporters on cable news cited as justifying and explaining the president’s counterstrike.

The criticism, as usual, seems to have had virtually no impact on the president. He kept up his attacks this weekend by telling an audience of veterans in Washington that the media failed to stop him from achieving his objectives.

“The fact is the press destroyed themselves because they went too far,” Mr. Trump said at an event at the Kennedy Center. “Instead of being subtle and smart, they used the hatchet and the people saw it right from the beginning.”

Mr. Trump, who has seized on an erroneous story by CNN in the past week to illustrate his point, said the media have been working against him from the start.

“My administration is transferring power outside of Washington and returning it to where it belongs: the people,” Mr. Trump said. “The fake media is trying to silence us, but we will not let them. The people know the truth. The fake media tried to stop us from going to the White House, but I’m president and they’re not.”

Hours after that speech, Mr. Trump posted the doctored WrestleMania footage, which CNN quickly condemned.

“It is a sad day when the president of the United States encourages violence against reporters,” the news network said in a statement. “Instead of preparing for his overseas trip, his first meeting with Vladimir Putin, dealing with North Korea and working on his health care bill, he is instead involved in juvenile behavior far below the dignity of his office. We will keep doing our jobs. He should start doing his.”

Administration officials on Sunday mostly tried to avoid discussing the president’s tweets.

Health and Human Services Secretary Tom Price skirted the issue and focused on health care reform. Marc Short, the White House director of legislative affairs, spoke only in general terms, saying Mr. Trump will use Twitter to get his message out to the people.

Thomas Bossert, the White House national security and counterterrorism adviser, went the furthest, saying Mr. Trump is “the most genuine” president in history. He added that no one should view the WrestleMania-CNN video as a threat.

“I think that no one would perceive that as a threat. I hope they don’t,” he told ABC News. “I do think that he’s beaten up in a way on cable platforms that he has a right to respond.”

What’s clear, however, is that Mr. Trump’s continued use of Twitter takes attention away from real legislative efforts, such as health care reform. Sunday’s cable news shows spent nearly as much time discussing Mr. Trump’s tweets as they did on the embattled Senate health care bill.

“I think in Washington we have bigger issues than people being outraged by somebody else’s tweet,” Mr. Kasich said.

• Dave Boyer contributed to this report.

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