WASHINGTON — After a half-day of animated debate, the Senate Finance Committee on Tuesday rejected efforts by liberal Democrats to add a government-run health insurance plan to major health care legislation, dealing the first official setback to an idea that many Democrats, including President Obama, say they support.

All of the other versions of the health care legislation advancing in Congress — a bill approved by the Senate health committee and a trio of bills in the House — include some version of the government-run plan, or public option.

But the Finance Committee chairman, Senator Max Baucus, Democrat of Montana, long ago removed it from his proposal because of stiff opposition from Republicans who call the public plan a step toward “socialized medicine.”

The committee on Tuesday afternoon voted, 15 to 8, to reject an amendment proposed by Senator John D. Rockefeller IV, Democrat of West Virginia, to add a public option called the Community Choice Health Plan, an outcome that underscored the lack of support for a government plan among many Democrats.

Mr. Baucus voted no, as did Senators Thomas R. Carper of Delaware, Kent Conrad of North Dakota, Blanche Lincoln of Arkansas, and Bill Nelson of Florida, joining all 10 Republicans in opposition.

A second amendment by Senator Charles E. Schumer, Democrat of New York, to create a different version of a public plan was also defeated, though by a closer margin, 13 to 10, with the added support of Mr. Carper and Mr. Nelson.

Mr. Schumer who voted in favor of both proposals, said supporters of the public option would keep on fighting.

“We are going to keep at this and at this and at this until we succeed, because we believe in it so strongly,” he said.

Advocates of a public plan say it would provide crucial competition for private insurers and that the larger goals of the legislation, to extend coverage to more than 30 million uninsured Americans and to slow the steep rise in health care costs, cannot be achieved without it.

The debate came as the Finance Committee resumed debate over the health care bill after a three-day weekend because of Yom Kippur, the Jewish Day of Atonement.

In the emotionally charged debate, Mr. Rockefeller railed against the practices of private insurers, who he suggested were largely preying on a defenseless American public. “They’re getting away with terrible things,” he said.

But Senator Charles E. Grassley of Iowa, the senior Republican on the Finance Committee, stepped in to voice his party’s fierce opposition to the idea of government-run insurance.

“A government run plan will ultimately force private insurers out of business,” Mr. Grassley said, adding that supporters of the public option were trying to open a back door toward a fully government-run, or single-payer, health system like those in Canada or England.

“Public option is a step toward a completely government run plan that they are hoping for,” Mr. Grassley said.

And he rejected assertions by Democrats, including Mr. Rockefeller, that the public plan would compete fairly because it would have to follow the same rules as private insurers.

“The federal government will not only be running the plan, it will also be running the market in which it competes with the private plans and that doesn’t sound like a level playing field to me,” Mr. Grassley said.

Democrats quickly rose up to answer the charges, including Mr. Schumer, who challenged Mr. Grassley to spell out his views on Medicare, the government insurance plan for Americans over age 65 and for the disabled.

“I just want to know what you think of Medicare, which is a much more government-run program,” Mr. Schumer said.

“I think that Medicare is part of the social fabric of America just like Social Security is,” Mr. Grassley said. “To say that I support it is not to say that it’s the best system that it could be.”

“But it is a government-run plan,” Mr. Schumer shot back.

Mr. Grassley, a veteran Senate debater, insisted that Medicare did not pose a threat to the private insurance industry. “It’s not easy to undo a Medicare plan without also hurting a lot of private initiatives that are coupled with it,” he said.

Mr. Schumer pounced. “You are supportive of Medicare,” he said. “I just don’t understand the difference. That’s a government-run plan and the main knock you have made on Senator Rockefeller’s amendment, and I am sure on mine, is that it’s government-run.”

The efforts by Mr. Rockefeller and Mr. Schumer to add a public plan to the bill were really just a dress rehearsal for a fuller battle that will play out on the Senate floor in the weeks ahead.

Senate Democratic leaders, however, do not believe there will be sufficient support to add the public option to the bill.

Aides to the majority leader, Senator Harry Reid, Democrat of Nevada, say that he will not include a provision for the public option when he combines the measures coming out of the finance and health committees.

Mr. Rockefeller, Mr. Schumer and other supporters of a government-run plan will bring floor amendments trying once again to add it to the legislation.

And even when the debate over the public option is taken up on the Senate floor, most likely it will not be finished.

There is wider support for a government-run insurance plan in the House, where the Democratic caucus is more liberal. And if the House bill includes a public option, as Speaker Nancy Pelosi has indicated, the issue will ultimately be decided in a conference proceedings to reconcile the Senate and House bills.

As an alternative, Mr. Baucus included in his bill a proposal to create private, nonprofit health insurance cooperatives to compete with private insurers.

The Congressional Budget Office has questioned whether the cooperatives would really have much effect. And there are Democrats and Republicans on the Finance Committee who have proposed amendments that would strip the cooperative provisions.

The main architect of that proposal, Mr. Conrad said during the committee debate that it would provide “strong not for profit competition to the private insurers.” But he warned that hospitals in his home state would be devastated by Mr. Rockefeller’s proposal, which would initially tie the public plan’s payment rates to the rates paid by Medicare.

Many hospitals, doctors and other health care providers say Medicare rates are too low.

Mr. Conrad urged his colleagues to consider his alternative,. “We have gotten locked in a very sterile debate ,” he said.