Facing growing outrage from Americans, President Barack Obama reversed course Thursday and offered to let insurance companies sell existing plans even if they don’t meet the minimum standards set by his new problem-fraught health care law.

But Obama’s much-delayed attempt to make good on his promise that Americans could keep their insurance plans if they liked them faces strong opposition from insurance companies, which warn that rates might spike, and it risks undermining the basic premise of his law, which requires quality, affordable insurance.

“We fumbled the rollout on this health care law,” a contrite Obama said in an hourlong news conference Thursday at the White House. “We should have done a better job getting that right on day one – not on day 28 or on day 40.”

The president flatly took responsibility for the slew of problems that have bedeviled his signature domestic achievement, including an error-filled website, and said he’d restore Americans’ confidence in him by fixing this month and making good on his promise to allow Americans to keep their plans. “We’re just gonna keep on chipping away at this,” he said.

The debacle threatens to swamp Obama’s entire second-term agenda, raising questions about his competency and credibility. Polls released this week show the president’s job-approval rating at a historic low and a majority of voters saying, for the first time, that he isn’t trustworthy.

“A White House interested in stabilizing this presidency would want to leave no stone unturned in the effort to deal with both those problems,” said William Galston, a former policy adviser to President Bill Clinton who’s a senior fellow at the Brookings Institution, a center-left policy research center.

On Thursday, Obama announced that he’d allow – but not require – insurance companies to extend existing policies for a year as long as they notified customers that their benefits might be diminished with their current plans and that alternative policies might be available to them.

Insurance companies already have devised plans for next year, received the necessary approval from states and begun to sell policies. They aren’t required to continue to offer their existing policies and state insurance commissioners aren’t required to approve those 2013 plans.

“Changing the rules after health plans have already met the requirements of the law could destabilize the market and result in higher premiums for consumers,” Karen Ignagni, the president and CEO of America’s Health Insurance Plans, which represents the industry.

Washington state’s insurance commissioner, Mike Kreidler, announced Thursday he won’t allow insurers to extend their policies, saying Washington’s state-based exchange was “up and running and successfully enrolling thousands of consumers.”

“We are staying the course,” said Kreidler, a Democrat who’s one of the longest-serving state insurance commissioners in the country.

California Insurance Commissioner Dave Jones said he’d taken steps to allow more than 1 million residents with terminating insurance plans to keep them.

Companies based their rates for 2014 on certain assumptions, including that consumers who bought plans after 2010 or had their prior plans substantially change wouldn’t be able to keep them. If more customers are allowed to keep their plans, companies probably will have to raise premiums and offer fewer choices.

“The point of the law was to provide health care insurance that was affordable and comprehensive and that protected people from catastrophic, out-of-pocket costs, so going back and allowing people to keep plans that are substandard seems not to fit the goal of the law,” said Sara Collins, the vice president for health care coverage and access at the Commonwealth Fund, a private foundation that promotes access to health care and has supported the Affordable Care Act.

The White House says it will try to keep prices down through an existing program that could provide companies financial assistance. But Obama, who was limited in what he could do without approval from an often unfriendly, inactive Congress, acknowledged that his administrative fix might not solve the problem and that lawmakers may still have to act.

The Republican-led House of Representatives and Democratic-controlled Senate are considering bills that would allow people to keep their plans. The House will vote Friday on legislation that would let new customers buy existing policies, not just current customers. The White House says the House bill goes too far, but some frustrated Democrats, many of whom face tough re-election campaigns next year, say Obama hasn’t done enough to fix the problems.

“I am highly skeptical that they can do this administratively,” House Speaker John Boehner, R-Ohio said at a news conference. “There is no way to fix this.”

In spite of the anxieties of some Democrats in the Senate and House, the party’s House leadership circled the wagons Thursday, offering to help fix the law’s problems but making no apologies for them.

“Our members were pleased with the president’s statement today,” said House Minority Leader Nancy Pelosi, D-Calif.

Sherry Taylor, 63, of Tulare, Calif., said she welcomed Obama’s news after her Anthem Blue Cross policy was canceled in lieu of a slightly higher-priced one. “I’m going to keep my policy until Congress works out what they’re actually going to do,” said the hairdresser, who broke her pelvis in two places in a bicycle accident.

Administration officials say they don’t know how many people would be affected by Obama’s decision, though they continue to insist that the problem is limited to people who buy their own insurance, about 11 million people. But a 2010 administration estimate indicates that as many as 69 percent of people with certain employer-based insurance plans – as many as 41 million – could have lost their policies.

Nicholas Bagley, a health policy expert and law assistant professor at the University of Michigan, said Obama’s fix raised more questions than answers, from the effect on premiums to the ability of insurance commissioners to swiftly approve previously canceled plans. And, he said, it remains to be seen whether the effort is legal.

“The administration hasn’t offered a legal justification, so it’s difficult to deliver a thumbs up or thumbs down,” he said. “There’s at least a big question mark whether the president can do this.”