Republicans said it was his fault, not theirs, and embarked on a strategy - opposed by Democrats - of voting on bills to reopen individual agencies or programs.
Ominously, there were suggestions from leaders in both parties that the shutdown, heading for its second day, could last for weeks and grow to encompass a possible default by the Treasury if Congress fails to raise the nation's debt ceiling. The two issues are "now all together," said Sen. Dick Durbin, D-Ill.
Speaking at the White House, the president accused Republicans of causing the first partial closure in 17 years as part of a non-stop "ideological crusade" to wipe out his signature health care law.
House Speaker John Boehner, R-Ohio, gave as good as he got. "The president isn't telling the whole story,' he said in an opinion article posted on the USA Today website. "The fact is that Washington Democrats have slammed the door on reopening the government by refusing to engage in bipartisan talks."
Both houses of Congress met in a Capitol closed to regular public tours, part of the impact of a partial shutdown that sent ripples of disruption outward - from museums and memorials in Washington to Yellowstone and other national parks and to tax auditors and federal offices serving Americans coast to coast.
Officials said roughly 800,000 federal employees would be affected by the shutdown after a half-day on the job Tuesday to fill out time cards, put new messages on their voice mail and similar chores.
Among those workers were some at the National Institute of Health's famed hospital of last resort, where officials said no new patients would be admitted for the duration of the shutdown. Dr. Francis Collins, agency director, estimated that each week the shutdown lasts will force the facility to turn away about 200 patients, 30 of them children, who want to enroll in studies of experimental treatments. Patients already at the hospital are permitted to stay.
Late Tuesday, House Republicans sought swift passage of legislation aimed at reopening small slices of the federal establishment. The bills covered the Department of Veterans Affairs, the Park Service and a portion of the Washington, D.C., government funded with local tax revenue.
Democrats generally opposed all three, saying Republicans shouldn't be permitted to choose which agencies remain open and which stay shut. As a result, all fell well short of the two-thirds majority needed for passage.
The White House also issued veto threats against the bills, drawing a jab from Michael Steel, a spokesman for Boehner. Obama "can't continue to complain about the impact of the government shutdown on veterans, visitors at National Parks, and D.C. while vetoing bills to help them," he said.
Several House Democrats used the occasion to seek a vote on a standalone spending bill, a measure that Rep. Elizabeth Esty of Connecticut said would "end the tea party shutdown." The requests were ruled out of order.
Republican aides said all three bills that were sidetracked could be brought up again on Wednesday under rules requiring a mere majority to pass. They said the House might also vote on a measure to reopen the hospital at the NIH, after several Democrats cited the impact on patients.
Ironically, a major expansion of the health care law - the very event Republicans had hoped to prevent - was unaffected as consumers flocked for the first time Tuesday to websites to shop for coverage sold by private companies.
The talk of joining the current fight - the Republicans are trying to sidetrack the health care law by holding up funding for the fiscal year that began at midnight Monday - to a dispute involving the national debt limit suggested the shutdown could go on for some time.
The administration says the ceiling must be raised by mid-month, and Republicans have long vowed to seek cuts in spending at the same time, a condition Obama has rejected.
In Washington, some Republicans conceded privately they might bear the brunt of any public anger over the shutdown - and seemed resigned to an eventual surrender in their latest bruising struggle with Obama.
Democrats have "all the leverage and we've got none," said Sen. Saxby Chambliss of Georgia.
Rep. Devin Nunes, R-Calif., said sardonically his party was following a "Ted Cruz-lemmings strategy" - a reference to the senator who is a prime proponent of action against the health care overhaul - and Rep. Scott Rigell of Virginia said it was time to pass legislation reopening the government without any health care impediments. "The shutdown is hurting my district - including the military and the hard-working men and women who have been furloughed due to the defense sequester," he said.
But that was far from the majority view among House Republicans, where tea party-aligned lawmakers prevailed more than a week ago on a reluctant leadership to link federal funding legislation to "Obamacare." In fact, some conservatives fretted the GOP had already given in too much.
Gone is the Republican demand for a full defunding of the health care law as the price for essential federal funding. Gone, too, are the demands for a one-year delay in the law, a permanent repeal of a medical device tax and a provision making it harder for women to obtain contraceptive coverage.
In place of those items, Republicans now seek a one-year-delay in the requirement for individuals to purchase insurance, and they want a separate provision that would dramatically raise the cost of health care for the president, vice president, members of Congress and thousands of aides.
Boehner has declined to say whether he would permit a vote on a stand-alone spending bill to reopen the government, stripped of health care provisions, though Democrats and Obama continued to call on him to do so. "He's afraid it will pass," said Durbin.
Sen. Cruz, R-Texas, the most prominent advocate of the "Defund Obamacare" movement, said the Senate should follow the House's lead and quickly reopen programs for veterans and the parks. Asked why it was appropriate to do so without demanding changes in the health care law, he offered no answer.
"None of us want to be in a shutdown. And we're here to say to the Senate Democrats, 'Come and talk to us,'" said House Majority Leader Eric Cantor, R-Va., as GOP lawmakers called for negotiations with the Senate on a compromise.
It was an offer that Senate Democrats chose to refuse, saying there was nothing to negotiate until Republicans agreed to reopen the federal establishment.
"The government is closed because of the irrationality of what's going on on the other side of the Capitol," said Senate Majority Leader Harry Reid.
In addition to "closed" signs and barricades springing up at the Lincoln Memorial and other tourist attractions, NASA and the Environmental Protection Agency were virtually shuttered, and Obama said veterans centers would be shut down.
Government workers classified as essential, such as air traffic controllers, Border Patrol agents and most food inspectors, remained on the job.
So, too, members of the military, whose pay was exempted from the shutdown in separate legislation Obama signed late Monday. Employees whose work is financed through fees, including those who issue passports and visas, also continued to work. The self-funded Postal Service remained in operation, and officials said the government will continue to pay Social Security benefits and Medicare and Medicaid fees to doctors on time.
Associated Press writers Lauran Neergaard, Alan Fram, Josh Lederman, Nedra Pickler, Seth Borenstein and Andrew Taylor contributed to this report.