Plenty of others have skated up to the edge of racism and anti-Semitism with no consequence. Representative Andy Biggs, Republican of Arizona who branded Ms. Omar an anti-Semite, raised eyebrows in 2016 when he charged that the Jewish philanthropist George Soros was buying a local election for county attorney. Representative Louie Gohmert, Republican of Texas, complained bitterly that last week’s anti-Semitism resolution had been watered down, just a few months after he accused Mr. Soros of collaborating with the Nazis during World War II. Patrick Gaspard, the president of Mr. Soros’s Open Society Foundations, demanded Mr. Gohmert apologize for the “disturbing and false anti-Semitic slur.”

[Read also: “Why Some Republicans Voted Against the Antibigotry Resolution.”]

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.