WASHINGTON — For President Obama, the fiscal deal passed by Congress on Tuesday finally ends four years of debate with Republicans about raising tax rates on the wealthy. But it seemed to reopen a debate within his party about the nature of his leadership and his skills as a negotiator.

While Mr. Obama got most of what he sought in the agreement, he found himself under withering criticism from some in his liberal base who accused him of caving in to Republicans by not taxing the rich more. Just as Speaker John A. Boehner has been under pressure from his right, Mr. Obama faces a virtual Tea Party of the left that sees his compromise as capitulation.

The main difference is that in the Obama era, the Democratic establishment has been less influenced, or intimidated, by the left than the Republican establishment has been by the right. Liberals have not mounted sustained primary challenges to take out wayward incumbents the way conservatives have. All but three Democrats voting in the Senate and 16 in the House supported the compromise on Tuesday, even as most House Republicans balked, giving Mr. Obama more room to operate than Mr. Boehner.

But the wave of grievance from liberal activists, labor leaders and economists suggested that the uneasy truce between Mr. Obama and his base that held through the campaign season had expired now that there was no longer a threat of a Mitt Romney victory. It also offered a harbinger of the president’s next four years.