WASHINGTON—President Barack Obama, in a retreat aimed at quelling the latest protest over his health-care law, on Thursday said insurers can extend by one year those policies they had canceled for failing to meet the law's requirements.

Mr. Obama accompanied the policy shift with an admission of personal responsibility for the botched launch of the program.

"We fumbled the rollout on this health-care law," Mr. Obama said in a televised White House news conference. "Am I going to have to do some work to rebuild confidence around some of our initiatives? Yeah."

He added that the troubles had hurt his party: "There is no doubt that our failure to roll out the [health law] smoothly has put a burden on Democrats."

The president's support for allowing the insurance plans to continue for existing policyholders essentially shifted responsibility for the cancellations to the insurers.