“It’s a team problem; the president has to own a piece of this,” said Newt Gingrich, the former House speaker and an informal adviser to Mr. Trump. “I may not be clever enough to understand this, but I don’t see how a Republican president deepening his fight with the Republican majority leader gets him very far.”

Though Mr. Trump has far outstripped Mr. McConnell in public fury, it was the senator who initiated the flare-up this week. Speaking on Monday at a Rotary Club event in Kentucky, he said Mr. Trump had harbored “excessive expectations” about the pace of progress at the Capitol.

The remarks, though hardly scorching, registered as a pointed message from the laconic Senate leader, who has strained all year to wrangle his narrow majority. Mr. McConnell has gotten little help from an often disengaged and policy-averse president whose habit of setting arbitrary (and often shifting) deadlines has frustrated senators for months.

On Wednesday, Mr. Trump grew animated in a phone call with Mr. McConnell, expressing disappointment in the “excessive expectations” message and the repeal stumble.

On Thursday, Mr. McConnell’s team showed little appetite for a protracted feud.

“As we’ve said, the leader has repeatedly spoken about the path forward on repealing and replacing Obamacare — on the Senate floor, at media availabilities and in Kentucky,” said Antonia Ferrier, Mr. McConnell’s spokeswoman, in an email, using language nearly identical to a statement issued the day before. “But if he has any new statements today, I’ll be sure to send them along.”

For Mr. McConnell — who helped engineer perhaps Mr. Trump’s most meaningful feat to date, the elevation of Justice Neil M. Gorsuch to the Supreme Court — the episode follows a health care defeat that has tested the unity of his conference.

After a seven-year push to unravel President Barack Obama’s signature domestic achievement, Republican senators saw three of their own doom the effort last month in a dramatic early-morning vote: Susan Collins of Maine, Lisa Murkowski of Alaska and John McCain of Arizona.