While Republicans did rebuke Mr. King and occasionally criticize President Trump’s language — such as when he said there were “very fine people on both sides” of the white supremacist rally in Charlottesville, Va. — rarely have they engaged in the kind of public soul-searching Democrats undertook last week. Now many Democrats worry they have set a new standard, creating a precedent that mandates a major response every time a member transgresses rules of rhetorical decorum that are ill-defined and subject to dispute.

“We have to be careful,” said Ms. Kelly, a member of the Congressional Black Caucus, which fought to blunt criticism of Ms. Omar. “She was elected just like I was, and she has the right to say what she believes, just as I do.”

Then there is the question of what happens if there is a next time. Some Jewish groups are already demanding that Ms. Omar be stripped of her seat on the House Foreign Affairs Committee — a step Democratic leaders have thus far been loath to take.

“That’s the real question: what if this happens again? What are we going to do?” asked Representative Josh Gottheimer, Democrat of New Jersey, who was among those pushing strongly for a resolution focused solely on anti-Semitism.