WASHINGTON — Barack Obama won the presidency by exploiting a political environment that devoured George W. Bush in a second term plagued by sinking credibility, failed legislative battles, fractured world relations and revolts inside his own party.

President Obama is now threatened by a similar toxic mix. The disastrous rollout of his health care law not only threatens the rest of his agenda but also raises questions about his competence in the same way that the Bush administration’s botched response to Hurricane Katrina undermined any semblance of Republican efficiency.

But unlike Mr. Bush, who faced confrontational but occasionally cooperative Democrats, Mr. Obama is battling a Republican opposition that has refused to open the door to any legislative fixes to the health care law and has blocked him at virtually every turn. A contrite-sounding Mr. Obama repeatedly blamed himself on Thursday for the failed health care rollout, which he acknowledged had thrust difficult burdens on his political allies and hurt Americans’ trust in him.

“It’s legitimate for them to expect me to have to win back some credibility on this health care law in particular and on a whole range of these issues in general,” Mr. Obama said. The president did not admit to misleading people about whether they could keep their insurance, but again expressed regret that his assurances turned out to be wrong.