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On Monday, campaign spokeswoman Andrea Saul said the pressure is on Mr. Obama to choose between calling it a tax or accepting it as a penalty — and therefore unconstitutional.

But one Republican strategist said that was playing “a cat-and-mouse game” and urged Mr. Romney to move on.

“The Romney campaign is overplaying its hand on the penalty/tax debate, given that polling shows roughly 40 percent of voters don’t even know the Supreme Court has ruled on Obamacare and given that there still exists a lot of low-information voters out there who don’t even have a clear understanding of what Obamacare entails and who it affects,” said Ford O’Connell, a political strategist who ran the McCain-Palin 2008 campaign’s national rural outreach program.

For congressional Republicans, there was no doubt about what the high court said.

“The Supreme Court has spoken. This law is a tax,” Senate Minority Leader Mitch McConnell, Kentucky Republican, said in a fiery speech on the Senate floor after the court announced its ruling. “The bill was sold to the American people on a deception.”

Most in the party seem to agree with Mr. McConnell.

“It is a tax. I don’t think there is any question about it,” said Bob Bennett, chairman of the Ohio Republican Party.

Still, Mr. Bennett said the issue of labels won’t damage Mr. Romney’s broader message for voters, which is a stern vow to repeal the entire health care law.

“Look, Obamacare is an abomination on the American taxpayer and those who require health care. I think if you ask anybody — Republicans and Democrats — there are pieces of that in there that are good that I think both Democrats and Republicans can support. The bill as a whole is a bad bill,” he said.

“I think, you know, Romney is OK. He says he wants to repeal Obamacare and start over. I think that is the right thing to say,” Mr. Bennett said.

Numerous Capitol Hill Republicans were miffed by the Romney campaign’s decision to go with the “penalty” label Monday, but they are plowing ahead with their own tax arguments.

The chairman of Congress‘ top tax-writing committee announced Tuesday that he would hold a hearing looking at the effects of the newly labeled tax.

“I strongly disagree with the court’s decision and its holding that Congress can tax Americans who choose not to purchase government-approved health insurance,” said Chairman Dave Camp, Michigan Republican. “This ruling sets a dangerous precedent with potentially enormous consequences.”