Senate Majority Leader Harry Reid decried Mr. Bush's action as a "heartless veto."
"Never has it been clearer how detached President Bush is from the priorities of the American people," Reid, D-Nev., said in a statement. "By vetoing a bipartisan bill to renew the successful Children's Health Insurance Program, President Bush is denying health care to millions of low-income kids in America."
Democratic congressional leaders said they may put off the override attempt for as long as two weeks to maximize pressure on Republican House members whose votes will be critical.
"The President should not be so heartless when it comes to the children of America," Washington State Democrat Maria Cantwell said on the floor of the Senate. "I know my colleagues here are working shoulder to shoulder, Democrats and Republicans, trying to stop the President's veto."
"We remain committed to making SCHIP into law - with or without the president's support," said the leader of the House, Speaker Nancy Pelosi, a Democrat, referring to the full name of the State Children's Health Insurance Program.
Eighteen Republicans joined Democrats in the Senate, enough to override Mr. Bush's veto. But this was not the case in the House, where despite sizable Republican backing, supporters of the bill are about two dozen votes short of a successful override.
Illinois Rep. Rahm Emmanuel talked to CBS News chief White House correspondent Jim Axelrod about the political advantage the Democratic party sees in this situation.
"So the President of the United States is asking 15 Republicans to stand with him on an argument about government-run health care… and deny American kids health care," Emmanuel said, "and yet, vote at the same time to give Iraq 190 billion dollars."
House Majority Leader Steny Hoyer, D-Md., said Democrats were imploring 15 House Republicans to switch positions but had received no agreements so far.
House Minority Whip Roy Blunt, R-Mo., said he was "absolutely confident" that the House would be able to sustain Mr. Bush's veto.
Senate Minority Whip Trent Lott, R-Miss., said Congress should be able to reach a compromise with Mr. Bush once he vetoes the bill. "We should not allow it to be expanded to higher and higher income levels, and to adults. This is about poor children," he said. "But we can work it out."
The White House sought little attention for Mr. Bush's action, with the president casting his veto behind closed doors without any fanfare or news coverage. He defended it later Wednesday during a budget speech.
"Poor kids first," Mr. Bush said. "Secondly, I believe in private medicine, not the federal government running the health care system."
But he seemed eager to avert a full-scale showdown over the difficult issue, offering that he is "more than willing" to negotiate with lawmakers "if they need a little more money in the bill to help us meet the objective of getting help for poor children."
The Democrats who control Congress, with significant support from Republicans, passed the legislation to add $35 billion over five years to allow an additional 4 million children into the program. It would be funded by raising the federal cigarette tax by 61 cents to $1 per pack.
The president argued that the bill was too costly, took the program too far from its original intent of helping the poor, and would entice people now covered in the private sector to switch to government coverage. He wants only a $5 billion increase in funding.
Mr. Bush argued that the congressional plan would be a move toward socialized medicine by expanding the program to higher-income families.
Democrats deny that, saying their goal is to cover more of the millions of uninsured children and noting that the bill provides financial incentives for states to cover their lowest-income children first. Of the over 43 million people nationwide who lack health insurance, 9 percent, or over 6 million, are under 18 years old.
It took Mr. Bush six years to veto his first bill, when he blocked expanded federal research using embryonic stem cells last summer. In May, he vetoed a spending bill that would have required troop withdrawals from Iraq. In June, he vetoed another bill to ease restraints on federally funded stem cell research.
Mr. Bush's four vetoes are far fewer than any of his recent predecessors. In fact, reports CBS News White House correspondent Mark Knoller, you have to go back to James Garfield in 1881 to find the last president who cast fewer vetoes than Mr. Bush. FDR cast the most vetoes: 635 during 12 years in office. Eight presidents never vetoed any legislation, among them both John Adams and Thomas Jefferson.
In the case of the health insurance program, the veto is a bit of a high-stakes gambit for Mr. Bush, pitting him against both the Democrats who have controlled both houses of Congress since January, but also many members of his own party and the public.
The Democratic Congressional Campaign Committee launched radio ads Monday attacking eight GOP House members who voted against the bill and face potentially tough re-election campaigns next year.
And Gerald McEntee, president of the American Federation of State, County and Municipal Employees union, said a coalition of liberal groups planned more than 200 events throughout the nation to highlight the issue.
The issue could also have reverberations on the presidential campaign trail. Democratic candidate John Edwards called it "shocking" that Republican candidates Fred Thompson, Rudy Giuliani, John McCain and Mitt Romney were "lining up with President Bush and against health care for our children."
Edwards said it's time for the Republicans "to start picking on someone their own size."