On Tuesday, uninsured Americans, about 15 percent of the population, could begin enrolling in the state-based insurance marketplaces, known as exchanges, created by the 2010 law.

It was unclear how long the shutdown might go on, and no one in either party could say if Republicans would remain firm in their insistence that Democrats agree to significant changes to the health care law. But on the initial day of the first shutdown in nearly 18 years — when a Republican-controlled Congress battled another Democratic president, Bill Clinton — there was little business getting done in the House or the Senate other than photo opportunities and partisan speeches.

One senior Republican, Representative Paul D. Ryan of Wisconsin, the House Budget Committee chairman, indicated that the stalemate could go on for at least two more weeks until the nation reaches its borrowing limit. He said the deadline to address the debt limit, and avoid a default, could be “the forcing mechanism to bring the two parties together.”

Yet House Republicans also vow to oppose an increase in the debt ceiling unless Mr. Obama delays the health care law.

On Tuesday night, Speaker John A. Boehner and House Republicans tried to ease the effects of the shutdown and force Democrats into negotiation. The Republicans proposed three bills — to finance veterans’ programs, the National Park Service and federally run services in Washington — but because they introduced the measures under a fast-track procedural rule that required a two-thirds vote, each of them failed.