WASHINGTON — For President Barack 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 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 House Speaker John Boehner has been under pressure from his right, 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 Obama more room to operate than Boehner.

But the wave of grievance from liberal activists, labor leaders and economists suggested that the uneasy truce between 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.

The criticism has irritated the White House, which argued that Obama held true to principle by forcing Republicans to raise income tax rates on the wealthy and extend unemployment benefits and targeted tax credits. Obama also quashed Republican demands to trim the growth of entitlement benefits. Aides dismissed armchair criticism from those who have never had to negotiate with intractable opposition.

“There’s some frustration that over time you would think everybody would have a better understanding of the parameters of this,” said Robert Gibbs, a longtime adviser to Obama who once called such critics “the professional left.” “But he understands now probably better than at any other point in his presidency what it means to be a leader, what it means to have to do things that are good not just for one party but good for the country.”

The criticism from the left mirrors past complaints when Obama included tax cuts in his stimulus package, gave up on a government-run option in health care negotiations and temporarily extended Bush-era tax cuts for the wealthy two years ago. Liberals said Obama should have capitalized on his re-election victory and the expiration on New Year’s Day of all of the Bush tax cuts to force Republicans to accept his terms.

“The president remains clueless about how to use leverage in a negotiation,” said Adam Green, a co-founder of the Progressive Change Campaign Committee, a liberal advocacy organization. “Republicans publicly admitted they lost the tax debate and would be forced to cave, yet the president just kept giving stuff away.”

Robert Reich, the former labor secretary, said that Obama “has stiffened his tactical resolve” but that “he’s still the same President Obama who wants a deal above all else and seems willing to compromise on even the most basic principle.”

Richard Trumka, the president of the AFL-CIO, said in a Twitter message on Monday that the agreement was “not a good fiscal cliff deal if it gives more tax cuts to 2 percent.” Sen. Tom Harkin, D-Iowa, said on the floor on Monday that “this looks like a very bad deal.”