WASHINGTON (Reuters) - President George W. Bush on Wednesday vetoed a measure to expand a popular children's health care program, launching the first in a series of major battles with Democrats over domestic spending.

The bill, which had bipartisan support in the U.S. Congress, would have provided an extra $35 billion over five years for the health program. Cigarette taxes would have been raised to fund the expansion from the current $25 billion level.

Democrats accused Bush of neglecting the needs of poor children and vowed to try to override the veto. The veto is also likely to anger a number in Bush's own Republican Party who fear the issue will hurt them in the 2008 congressional and presidential elections.

But Bush said the funding level would have expanded the health program beyond its original intent and taken a step toward government-run health care.

"The policies of the government ought to be to help poor children and to focus on poor children. And the policies of the government ought to be to help people find private insurance, not federal coverage," Bush told a business forum in Lancaster, Pennsylvania.

"I happen to believe that what you're seeing when you expand eligibility for federal programs is the desire by some in Washington, D.C. to federalize health care. I don't think that's good for the country," he added.

Senate Democratic Leader Harry Reid of Nevada called the veto "heartless" and said it showed Bush had "turned his back on America's children."

Supporters of the bill said the additional $35 billion would have helped provide health coverage for some 10 million children.

Bush had proposed a much smaller increase of $5 billion over five years, a level that critics said would not even be enough to cover the current number of children in the program because of the rising costs of health care.

Bush said he was willing to sit down with Democrats to negotiate the funding.

His veto came as a Washington Post-ABC News poll showed more than seven in 10 Americans supported the $35 billion increase proposed under the bill. By contrast, the same poll showed many wanted to see a reduction in Bush's spending proposal for the Iraq war.

"Today the president showed the nation his true priorities: $700 billion for a war in Iraq, but no health care for low-income kids," said Rep. Rahm Emanuel, a Democrat of Illinois.

"Millions of American children and their families won't forget that they are on the bottom of the president's priority list," Emanuel said.

The Senate overwhelmingly backed the health legislation. But, the margin of support in the House of Representatives fell short of the two-thirds majority that would be needed to override a presidential veto. Democrats are trying to encourage Republicans to switch their votes.

The Democratic Party plans a series of television ads attacking Republicans over Bush's health care veto.

Bush, with 16 months left in his presidency, has also threatened to veto a series of annual funding bills to keep domestic spending within his proposed limit of $933 billion.

The president is aiming to cast Democrats as fiscally irresponsible as he tries to shore up support from conservatives, many of whom are angry at Bush for allowing big spending increases during his first six years in office.

The rejection of the health bill marks the fourth veto for Bush since he took office in 2001. He twice rejected legislation on stem cell research and also vetoed an Iraq war supplemental spending bill because it included timelines for withdrawing troops.

(Additional reporting by Tabassum Zakaria and Donna Smith)