Only four Republicans and one independent broke and voted with the Democrats to condemn the president’s language in the House vote Tuesday, a stark reminder of just how far the party has come from the period when its leaders believed their political future depended on being a big tent, welcoming to Latino and African-American voters.

Instead, a range of party leaders were pushing messages of patriotism. Some attempted to sidestep the racial implications, while others seemed less concerned about the potential blowback.

“Forget these four,” said Kellyanne Conway, the counselor to the president, in an interview with Fox News on Tuesday. “They represent a dark underbelly of people in this country,” she added. “We are sick and tired of people denigrating that American flag, the American military, veterans and America.”

Others were jumping on the bandwagon, but seeking to reframe and soften the message. Representative Liz Cheney of Wyoming, the third-ranking House Republican, effectively offered Mr. Trump a tutorial in how to go on the offensive without inviting a backlash.

“Our opposition to our socialist colleagues has absolutely nothing to do with their gender, with their religion or with their race,” Ms. Cheney told reporters Tuesday. “It has to do with the content of their policies.”

The election is still more than 15 months away, and eventually the Democrats will have a standard-bearer to define the party in opposition to Mr. Trump. Still, some Democrats worry that criticism of the four congresswomen will resonate with a segment of their voters and independents, who may prove just as uneasy with the policies they support, and some of their rhetoric, as they are with Mr. Trump’s own bombast.

The Democrats who fared the best in the midterms were those who played down Mr. Trump while highlighting issues like protecting the health insurance of people with pre-existing conditions. And many of the strategists who are rallying behind former Vice President Joseph R. Biden Jr. believe the party can’t count on increasing turnout among young people and minorities, and needs to lure back voters it lost to Mr. Trump.