Sen. Ted Cruz conceded Sunday that his strategy to leverage a government shutdown as a way to defund Obamacare would fail unless Republicans are able to convince voters that President Obama is to blame for cutting off federal programs.

The Texas Republican said on CNN that he "is not at all convinced" that Obama would refuse to defund the Affordable Health Care Act, his signature legislative achievement since winning the presidency in 2008. But Cruz, a potential 2016 GOP presidential candidate, seemed to indicate that supporters of his defund-or-shutdown Obamacare strategy face a steep political challenge in their campaign to convince voters to back their effort.

"President Obama [and Senate Majority Leader] Harry Reid [D-Nev.] will scream and holler that the mean, nasty Republicans are threatening to shut down the government, and at that point Republicans have to do something we haven't done in a long time: stand up and win the argument," Cruz said.

"We have to stand up and say, 'No, we want to keep the government open; we have voted to keep the government open, to fund the government.' Why is President Obama threatening to shut the government down to force Obamacare down the throats of the American people?"

Cruz said the strategy would only work if House Republicans are on board. Polling indicates that while the public, regardless of party, is wary of Obamacare they also oppose shutting down the government to defund it.

"The House of Representatives should pass a continuing resolution that funds the federal government in its entirety — every aspect of the federal government except Obamacare, and it should explicitly prohibit any funding for Obamacare, mandatory or discretionary," he said.

Howard Dean, a physician and former Democratic National Committee chairman, opposed Obamacare when it was first enacted, but defended it on Sunday, saying on CNN that "It's going to help a lot of people."

"Ted Cruz may be a very good politician, but he certainly doesn't know anything about health care," Dean said. "He's a slick spokesman, and God help us if he ever does get to be anything other than a senator from Texas."

But GOP conservatives are pressuring party leaders to defund Obamacare and support threatening to shut down the government unless it is. Heritage Foundation President Jim DeMint said that any Republican who doesn't support Cruz should be thrown out of office.

DeMint, a former South Carolina senator, has been traveling the country with Cruz trying to drum up support for the defund-or-shutdown strategy, which most congressional Republicans — conservatives included, still appear to oppose the idea.

"The Heritage Foundation doesn't get involved with elections," DeMint said Sunday on CNN. "But I have said on several occasions that if someone runs for office making a promise, such as many did, to stop Obamacare, and then they say they're afraid to do that because they might lose the next election, my personal opinion is they should be replaced."

Cruz said he has no plans to get involved in Republican primary campaigns in the 2014 election cycle, either for or against Democratic or Republican incumbents.

"I think it is likely that I'll stay out of all incumbent races, on either side," he said.