Republicans piled criticism on Barack Obama on Tuesday for what they described as an ideological inauguration speech, and accused him of failing to offer anything in the way of compromise.

Obama and his vice-president Joe Biden attended a service at the National Cathedral in Washington in the morning, marking the traditional end to four days of inaugural events.

The pastor, the Rev Adam Hamilton, called for the parties to work together. "We're in need of a new common national vision – not one that is solely Democratic, or solely Republican," he said.

But before Obama had even left the church, the hard reality of Washington politics began to kick in.

Republican Congressman Pete King said Obama could at least have made an effort to reach out to Republicans. "I thought he could've found some way to be more constructive," King told CNN, adding: "You can still make a case but do it in a more magnanimous way."

Senator John McCain, Obama's presidential opponent in 2008, rounded on Obama, saying it was the eighth inauguration speech he had heard and the first in which a president had failed to reach out his hand to the opposition party.

Obama devoted much of his first inaugural speech in 2009 offering to work with the Republicans but spent most of the intervening years in bitter, debilitating fights with them that have frequently left Washington paralysed. Obama, frustrated by that and emboldened by his election victory, used his inaugural speech to underline his new combative approach, one in which he outlined a progressive agenda that he hoped to push through in spite of Republican opposition.

Conservative commentators joined Republican members of Congress in criticising Obama. Stephen Hayes at the Weekly Standard accused him seeking "to move the country even further left".

His colleague Paul Kengor echoed that, saying: "In the hands of a conservative like Ronald Reagan, this would be just fine, a return to our founding principles. Yet, in the hands of a radical-left 'progressive' like Barack Obama, these are not the soothing words they appear to be on the surface."

The first reward for Obama's new hardline approach could come as early as Wednesday. The Republicans had threatened to close down Washington in a new economic showdown but are backtracking.

They had given Obama an ultimatum: to make deep spending cuts or they would refuse to vote through raising the debt limit, which could have meant department after department closing down and no cheques for veterans or welfare recipients. Obama, facing them down, told them to go ahead: let the federal government close down and take the consequences.

Instead, Republicans are scheduled to vote on Wednesday in the House, where they have a majority, on a bill to extend the debt limit for a further three months.

The White House said that, while it had some concerns about the bill, if the House and Senate passed it, Obama would sign it.

Obama's press secretary Jay Carney, described the debt ceiling bill as a "very significant development" in scaling down the sense of conflict and the potential for events spinning out of control.