WASHINGTON — In backing President Bush’s veto of a children’s health bill, many Republicans feel their party has picked the wrong issue to try to regain its long-lost reputation as guardian of prudent federal spending.

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 yesterday 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. Party leaders say the House will sustain Mr. Bush’s veto in two weeks.

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.

Senator Coburn, Republican of Oklahoma, 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,” Mr. 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,” Senator Hatch, Republican of Utah, told reporters yesterday. “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 Mr. Hatch’s colleagues say.

But Rep. Tom Feeney, Republican of Florida, 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,” Mr. Feeney, a target of Democratic radio ads attacking his support of the veto, said.