The move left Ms. Pelosi’s natural allies in the House’s Hispanic and progressive caucuses stunned and feeling betrayed. “Since when did the Problem Solvers Caucus become the Child Abuse Caucus?” Mr. Pocan said on Twitter. Representatives Max Rose, Democrat of New York, and Dean Phillips, Democrat of Minnesota, were later seen angrily confronting him on the House floor.

Representative Raúl M. Grijalva, Democrat of Arizona and a former co-chairman of the Congressional Progressive Caucus, said the episode left him “very resentful,” and feeling as if the moderates had essentially forced the House to silence its natural inclinations.

“I just hope that in forcing us to do nothing, they don’t feel like they’ve actually accomplished anything,” Mr. Grijalva said. “I don’t know what the motivation was, to try to assert some power or what, but in the future, we shouldn’t hesitate bringing our agenda and our legislation forward because it might offend 23 or 24” centrists.

The episode also exposed divisions among the moderates themselves. On the House floor on Thursday, about 10 moderate freshman Democrats huddled near the marble dais, arguing about the way forward. One lawmaker said if they sided with the other party in a bid to force the House to consider the weaker Senate bill, “we might as well be Republicans,” according to one person familiar with the exchange who described it on the condition of anonymity.

Representative Abigail Spanberger, Democrat of Virginia, grew red-faced and emotional during the exchange, and stormed off the House floor, returning a short time later and accepting an embrace from Representative Katie Porter, Democrat of California. Both ultimately supported the bill.

It was a version of a point that had been made, in much gentler fashion, during floor remarks by Representative Jim McGovern, Democrat of Massachusetts and the chairman of the Rules Committee. He warned that the procedural move the moderates were threatening to join “is a vote to give control of the House floor to the Republicans.”

But the moderates said they had done the party a favor, getting the House to the only tenable position as quickly as possible.

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