The shutdown "isn't about deficits, spending or budgets," a visibly angry Obama said in a statement in the Rose Garden. "This shutdown is about rolling back our efforts to provide health insurance to folks who don't have it."
"This, more than anything else seems to be what the Republican Party stands for these days," Obama said.
But, he added, "shutting down government doesn't accomplish their stated goal."
Up and running Tuesday -- albeit with glitches, Obama noted -- was the healthcare insurance marketplace where individuals can comparison-shop for health coverage. The exchanges -- available online, over the phone and in person -- are designed mainly for people who are uninsured or who buy insurance on their own because they don't get it through work.
"The Affordable Care Act is a law that passed the House, it passed the Senate. The Supreme Court ruled it constitutional. It was a central issue in last year's election. It is settled, and it is here to stay," he said. "And because of its funding sources, it's not impacted by a government shutdown."
The impact of the shutdown that began at midnight may not be known for a while, but the last time Republicans shuttered the federal government in 1996, "it hurt our economy," Obama said.
Even though he signed legislation to ensure active military personnel will be paid, "hundreds of thousands of civilian employees" won't be getting paychecks, the president said.
"I urge the House Republicans to reopen the government and restart the services the American people depend on and allow public servants who have been sent home to return to work," Obama said. "It is only going to happen when Republicans realize they don't get to hold the entire economy hostage over ideological demands."
He repeated his position he would work with Democrats and Republicans "to do the things we need to do -- to grow the economy and create jobs and get our fiscal house in order over the long run."
He also repeated another pledge: He won't negotiate over raising the debt ceiling in mid-October.
"I will not negotiate over Congress' responsibility to pay bills it's already racked up," Obama said.
"The last time Republicans even threatened this course of action, many of you remember back in 2011. Our economy staggered. Our credit rating was downgraded for the first time," Obama said. "If they go through with it this time and force the United States to default on its obligations for the first time in history, it'd be far more dangerous than a government shutdown, as bad as a shutdown is. It would be an economic shutdown."
"I'm not going to allow anybody to drag the good name of the United States of America through the mud just to refight a settled election or extract ideological demands," he said. "Nobody gets to hurt our economy and millions of hard-working families over a law you don't like."
Throughout Monday and early Tuesday, the GOP-controlled House passed -- and the Democratically controlled Senate rejected -- measures to avert a shutdown, but with Obamacare-related strings attached. The Senate leadership said it wanted a clean stopgap funding bill.
"The government is closed," Senate Majority Leader Harry Reid said. "All over America, federal employees are getting furloughs this morning ... because of the irrationality that is going on in the other side [the House] of the Capitol."
Senate Minority Leader Mitch McConnell accused Democrats of rooting for a shutdown all along.
"Democratic leaders in Congress finally have their prize -- a government shutdown that no one seems to want but them," McConnell said. "With just hours left to go Democrats voted again and again to reject reasonable legislation."
"They don't even want to talk about it."
The shutdown means 800,000 of the government's 2.9 million federal workers are furloughed and more than a million others have been asked to work without pay.
The Office of Management and Budget issued orders just before the midnight deadline for agencies to "execute plans for an orderly shutdown due to the absence of appropriations" from Congress.
As part of the orderly shutdown, many federal workers will report to their agencies for a half day of shutdown preparations before being sent home.
Essential functions such as law enforcement and air-traffic control continue, although tens of thousands of controllers, prison guards and Border Patrol agents are required to work without pay.
National parks, monuments and museums, as well as most federal offices, are closed. Many federal activities, including Internal Revenue Service audits and flu-outbreak surveillance, have been suspended.
Numerous congressional hearings -- including one scheduled for Tuesday on last month's Washington Navy Yard shootings -- are postponed.
Celebrity Breakups and divorces of 2014 [PHOTOS]