A delay on Mr. Rosenstein’s appointment would create a number of problems for the Justice Department. In particular, he was expected to oversee any department investigations into Russia’s meddling in the presidential election after Mr. Sessions recused himself because Mr. Sessions was an adviser to the Trump presidential campaign.

Previous presidents have faced similar uprisings within their own parties: Democratic lawmakers initially denounced President Bill Clinton’s behavior with an intern that led to his impeachment in the House, and Republican frustration with the Iraq war, as President George W. Bush’s approval ratings fell, hampered Mr. Bush’s second-term agenda.

But rarely does a president clash so forcefully with his own party so early in his first year. Mr. Trump already faces a difficult dynamic on Capitol Hill as he struggles to push through a major overhaul of the nation’s health care system that is already dividing the Republican-controlled Congress.

To overcome that intraparty opposition — not to mention the hostility to his health care plan from Democrats — Mr. Trump will need to woo the very Republicans who are increasingly growing weary of defending his online assertions.

In the days since Mr. Trump’s Twitter post on March 4, the White House has offered a series of shifting response, explanations and clarifications, some of which have been in conflict with each other.

Democrats have been particularly aggressive in assailing the president and his staff. Representative Adam B. Schiff of California, the House Intelligence Committee’s ranking Democrat, said Wednesday that if no evidence emerged to substantiate his claim, Mr. Trump should “explain himself.”

“You can’t level an accusation of that type without retracting it or explaining just why it was done,” he said.