The standoff continued after a White House summit with chief executives as financial leaders and Wall street urged a resolution before serious damage is done to the U.S. and world economy.
Obama "refuses to negotiate," House Speaker John Boehner, R-Ohio., told reporters after private talks that lasted more than an hour. "All we're asking for here is a discussion and fairness for the American people under Obamacare."
But Senate Majority Leader Harry Reid of Nevada said moments later, "We're locked in tight on Obamacare" and neither the president nor Democrats will accept changes in the nation's 3-year-old health care law as the price for spending legislation needed to end the two-day partial shutdown.
With the nation's ability to borrow money soon to lapse, Republicans and Democrats alike said the shutdown could last for two weeks or more, and soon oblige a divided government to grapple with both economy-threatening issues at the same time.
The White House said in a statement after the meeting that Obama had made it clear "he is not going to negotiate over the need for Congress to act to reopen the government or to raise the debt limit to pay the bills Congress has already incurred."
It added, "The president remains hopeful that common sense will prevail."
The high-level bickering at microphones set up outside the White House reflected the day's proceedings in the Capitol.
The Republican-controlled House approved legislation to reopen the nation's parks and the National Institutes of Health, even though many Democrats criticized them as part of a piecemeal approach that fell far short of what was needed. The bills face dim prospects in the Senate, and the White House threatened to veto both in the unlikely event they make it to Obama's desk.
"What we're trying to do is to get the government open as quickly as possible," said the House majority leader, Rep. Eric Cantor of Virginia. "And all that it would take is us realizing we have a lot in agreement."
Earlier, an attempt by Democrats to force shutdown-ending legislation to the House floor failed on a 227-197 vote, with all Republicans in opposition. That left intact the tea party-driven strategy of demanding changes to the nation's health care overhaul as the price for essential federal financing, despite grumbling from Republican moderates.
The stock market ended lower as Wall Street CEOs, Europe's central banker and traders pressed for a solution. Chief executives from the nation's biggest financial firms met Obama for more than an hour Wednesday, some of them plainly frustrated with the tactics at play in Congress and with the potential showdown coming over the debt limit.
"You can re-litigate these policy issues in a political forum, but we shouldn't use threats of causing the U.S. to fail on its obligations to repay its debt as a cudgel," Lloyd Blankfein, CEO of Goldman Sachs, said after the meeting.
Democrats were scathing in their criticism.
"The American people would get better government out of Monkey Island at the local zoo than we're giving them today," said Rep. John Dingell of Michigan.
The Republican National Committee announced it would pay for personnel needed to reopen the World War II Memorial, a draw for aging veterans from around the country that is among the sites shuttered. In a statement, party chairman Reince Priebus challenged Democrats "to join with us in keeping this memorial open."
Democrats labeled that a stunt. "We've already been working on a plan to open the Memorial — and the entire government — after the GOP caused them to close," said party spokesman Mo Elleithee. "It's called a clean" spending bill.
As it turned out, more than 125 World War II veterans from Mississippi and Iowa who were initially kept out of the memorial Tuesday were escorted to the site with the help of members of Congress. Officials made further arrangements to allow veterans groups into the memorial during the shutdown.
A sampling of federal agencies showed how unevenly the shutdown was felt across the government.
The Environmental Protection Agency and Department of Housing and Urban Development listed only six per cent of their employees as essential, and therefore permitted to work during the impasse. James R. Clapper, director of national intelligence, said about 70 per cent of civilian employees in agencies under his control had been sent home.
By contrast, about 86 per cent of employees of the Department of Homeland Security remained on the job, and 95 per cent at the Veterans Affairs Department.
One furloughed employee, meteorologist Amy Fritz, said, "I want to get back to work." At a news conference arranged by congressional Democrats, the 38-year-old National Weather Service employee said she has more than $100,000 in student loan debt and is looking at ways to cut her budget.
In an interview with CNBC before meeting with lawmakers, Obama said he would not negotiate with Republicans until the government is reopened and Congress votes to raise the debt limit.
"If we get in the habit where a few folks, an extremist wing of one party, whether it's Democrat or Republican, are allowed to extort concessions based on a threat (to) undermine the full faith and credit of the United States, then any president who comes after me, not just me, will find themselves unable to govern effectively," he said.
"The White House said Obama would have to truncate a long-planned trip to Asia, calling off the final two stops in Malaysia and the Philippines.
The shutdown also intruded into the race for governor of Virginia.
Terry McAuliffe, the Democrat, said he supported legislation to guarantee retroactive pay for furloughed federal employees. The Republican contender, Ken Cuccinelli, called on members of Congress to decline their pay as long as the shutdown lasts.
The House sidetracked legislation Tuesday night to reopen some veterans programs, the national parks and a portion of the Washington, D.C., municipal government. All three bills fell short of the two-thirds majority needed when Democrats voted overwhelmingly against this.
Republicans tried again, this time under rules requiring only a simple majority. The parks measure was approved on a vote of 252-173, with 23 Democrats breaking ranks and voting in favour. The vote to reopen NIH was 254-171. The House also voted to allow the Washington, D.C., government to use the taxes it collects to operate programs.
Votes were deferred on more bills, one to assure pay for members of the National Guard and Reserves and another to allow some veterans programs to resume.
The NIH bill was added to the day's agenda after Democrats had said seriously ill patients would be turned away from the facility's hospital of last resort, and no new enrolment permitted in experimental treatments.
Democratic Rep. Louise Slaughter of New York said the Republican response was a ploy. "Every time they see a bad headline they're going to bring a bill to the floor and make it go away," she said.
Some Republicans took obvious pleasure in the rough rollout Tuesday of new health insurance markets created under Obama's health care law. Widespread online glitches prevented many people from signing up for coverage that begins in January.
Rep. Trey Radel of Florida said a 14-year-old could build a better website "in an afternoon in his basement."
At issue is the need to pass a temporary funding bill to keep the government open since the start of the new budget year on Tuesday.
Congress has passed more than 100 temporary funding bills since the last shutdown in 1996, almost all of them without controversy. The streak was broken because conservative Republicans have held up the current measure in the longshot hope of derailing or delaying Obamacare, just as the health insurance markets at the heart of the law opened on Tuesday.
Associated Press writers Andrew Taylor, Donna Cassata, Henry Jackson, Julie Pace, Jessica Gresko, Darlene Superville and Seth Borenstein contributed to this story
Also on HuffPost