Democrats gleefully concur and are pouring money, time and energy into efforts to make GOP leaders pay dearly for the decision.
Mr. Bush and most congressional Republicans say they support an expansion of the State Children's Health Insurance Program, known as SCHIP. But they want something considerably smaller than the $35 billion, five-year increase approved by the Democratic-led House and Senate and vetoed Wednesday by Mr. Bush.
Five dozen congressional Republicans supported the bill, which would significantly expand subsidized health insurance for children in families earning two or three times the federal poverty rate. Most Republicans opposed it, mainly because of its cost and size.
The argument now is whether the Democrats can get enough votes to override Mr. Bush's veto, reports CBS News senior White House correspondent Bill Plante. The Senate is no problem. But in the House, they are about 15 votes shy.
"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, and yet vote at the same time to give Iraq $190 billion," said Rep. Rahm Emmanuel, D-Ill.
That's how the Democrats are going to frame the argument. And because they control the scheduling, they're going to stretch this out for two weeks before they vote. Look for a lot of Republicans to be joining in trying to get this bill into law without the president's signature, says Plante.
The events have brought a long-simmering GOP debate to a full boil. Some Republicans feel their party was foolish to let spending and deficits soar while Mr. Bush was president and Republicans controlled Congress for a dozen years.
Sen. Tom Coburn, R-Okla., said in a recent Senate speech: "This will be the sixth time since 1997 that the debt limit has been raised."
"There is no system of economic controls," Coburn said. "My own party did a lot to create this mess."
Most Republican lawmakers have backed Mr. Bush in arguing that tax cuts and heavy spending on the Iraq and Afghanistan wars are more important than cutting federal spending overall. The deficit, as a share of the economy, they say, is not a huge problem.
Mr. Bush and his allies drew the line on the proposed SCHIP expansion, saying it would subsidize middle-income families that can afford private insurance. "Poor kids first," Mr. Bush said after vetoing the bill.
Mr. Bush's decision baffles and angers some Republican lawmakers who say the administration should have picked a less sympathetic program for an all-out fight with Democrats.
""He has been given advice that this is socialized medicine. Hardly," Sen. Orrin Hatch, R-Utah, told reporters Wednesday. "I hope the folks at home raise Cain."
Such positions are precisely why so many voters have grown disenchanted with the Republican Party, a number of Hatch's colleagues say.
But Rep. Tom Feeney, R-Fla., said in an interview that by aggressively defending and explaining Mr. Bush's veto, "there is an opportunity in the next few months for the Republicans to regain their brand."
"We can't win elections nationally if more Americans think Democrats are more fiscally responsible than Republicans," said Feeney, a target of Democratic radio ads attacking his support of the veto.
Polls show that voters see Democrats as better custodians of spending and fiscal affairs. A recent ABC News-Washington Post poll found an overwhelming preference for Democrats over Republicans - by 52 percent to 29 percent - when people were asked which party they trusted to do a better job handling the federal budget deficit.
A number of Republican lawmakers say their party has frittered away an important element of its heritage and appeal.
"Spending has been too high under Republican control and under Democratic control," Rep. Steve Chabot, R-Ohio, said in an interview. Like Feeney, he voted against the $35 billion SCHIP expansion and against the 2003 Medicare prescription drug benefit, which Mr. Bush embraced.
"This is another example of out-of-control spending," Chabot said of the children's health insurance bill.
Drawing the line on spending on the SCHIP bill, while the Iraq war rages on, is a matter of too little, too late, with too little explanation to voters, say Republicans who oppose Mr. Bush's veto.
"It's a hard bill to explain," especially for those who oppose it, said Rep. Deborah Pryce of Ohio, one of 45 House Republicans who voted for the $35 billion expansion. "It's partly the president's fault," she said, for doing too little to make his case to the public.
"People still congratulate me for my vote," Pryce said.
Bill Miller, a former House GOP aide who now is the U.S. Chamber of Commerce's political director, said Republicans have brought the dilemma on themselves.
Mr. Bush, he noted, "never vetoed any of those spending bills during the first six years of his presidency."
He said polls that consistently show voters feel Democrats are better at holding down spending "speaks volumes."
Republicans "are trying to figure out ways to reinstate the belief that they are a more fiscally responsible party than the Democrats," Miller added.
Democrats plan to spend months and millions of dollars telling voters that scaling back an expansion of the SCHIP program is hardly the way for Republicans to rediscover their old groove.