The insurers who were there, from more than a dozen of the nation’s largest companies, said that the discussion covered a wide range of issues, but that the president agreed his priority was to fix the health care website.

The insurance representatives said they would work with the administration to protect the financial viability of the new marketplaces, but did not say how they would do that. Afterward, Karen Ignagni, the president of America’s Health Insurance Plans, a trade group, said it was a “very productive” meeting, but would not go into detail.

The legislation approved by the House would go further than the fix announced on Thursday by Mr. Obama, who said he would temporarily waive some requirements of the law and allow insurers to renew “current policies for current enrollees” for a year.

Many of the Democrats who supported the bill are facing tough re-election fights back home, and expressed deep frustration with how the administration had handled the early implementation of Mr. Obama’s signature health care law. Representative Nick J. Rahall II, Democrat of West Virginia, who voted for the legislation, said that the White House deserved an “F-minus” for its botched rollout of the Affordable Care Act.

“I’m disgusted about it,” Mr. Rahall said. “I think heads should roll downtown. Whoever was responsible or may have known that this was going to occur should no longer be employed.”

Representative Ron Barber, Democrat of Arizona, who also joined Republicans in voting for the bill, was equally scathing, calling the rollout “a disaster.”

“My constituents are pretty upset,” he said, “and so am I.”

Representative Fred Upton, Republican of Michigan and the chief sponsor of the House bill, said it would fulfill a promise that Mr. Obama had made to the American people and then broken.

“In the last three years,” Mr. Upton said, “the president personally promised that if people liked their current health care plan, they could keep it ‘no matter what.’ But cancellation notices are now arriving in millions of mailboxes across the country. It’s cancellation today, sticker shock tomorrow.”

Mr. Upton, the chairman of the Energy and Commerce Committee, belittled Mr. Obama’s proposal, saying it was offered at the last minute, “as the administration’s allies in Congress panicked.”

Senior Democrats criticized the Upton legislation as a ploy that could unravel the entire health care law.

“Don’t pretend you care about the American people’s health care here,” said Representative Mike Doyle, Democrat of Pennsylvania. “You just want to repeal the Affordable Care Act. Democrats are not going to let you do that.”

The outlook for the legislation is unclear in the Senate, where Democrats running for re-election in 2014 are looking for a way to help consumers facing the loss of insurance policies that do not meet requirements of the 2010 law.

Senator Mary L. Landrieu, Democrat of Louisiana, was one of the first Democrats to break with the White House and offer her own proposal, which would allow people to keep their current plans indefinitely.

However, after the president’s turnabout on Thursday, many Senate Democrats said they were waiting to see if additional legislation was necessary, and quick action in the Senate is not expected.

House Democrats on Friday used a procedural maneuver to offer a plan of their own called “Landrieu lite,” intended to build on the president’s fix and offer their members additional political cover.

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