“Frankly, this isn’t what the American people elected them to do,” Representative Steve Scalise of Louisiana, the second-ranking Republican in the House, said on Wednesday night on Fox News. “I think there is going to be a heavy price for Democrats who go down this path.”

Mr. Mueller’s statement on Wednesday, while carefully tracking his written report, shook up the debate in Washington and clearly got under the president’s skin. Mr. Mueller reported that he could not establish any illegal conspiracy between Mr. Trump’s campaign and Russia, but he said he could not clear the president of obstruction of justice and essentially suggested that Congress take up that question.

Mr. Trump fumed with friends that Mr. Mueller was out to get him. Bill O’Reilly, the former Fox News host, told WABC-AM radio that Mr. Trump called him at 11 that night and complained about Mr. Mueller’s supposed conflicts.

The president’s case for his own defense, however, was marred by his own contradictory statements on Thursday as he swung back and forth in his characterizations of Russia’s actions during the 2016 presidential election.

“I had nothing to do with Russia helping me to get elected,” Mr. Trump wrote on Twitter before leaving the White House in the morning. While denying any involvement, the president’s comment was an acknowledgment that Moscow not only intervened in the campaign but also did so for the express purpose of helping elect him — the consensus conclusion of the nation’s intelligence agencies.

But barely a half-hour later, Mr. Trump emerged on the South Lawn and took it back. “No, Russia did not help me get elected,” he told reporters as he prepared to depart for Colorado Springs. “You know who got me elected? I got me elected. Russia didn’t help me at all. Russia, if anything, I think, helped the other side.”

In fact, even President Vladimir V. Putin of Russia has said publicly that he favored Mr. Trump over Hillary Clinton in 2016, although he has denied orchestrating an influence campaign on behalf of the Republican candidate.