Heritage Action CEO Michael Needham said the group would only support a plan to reopen the government if it also defunds ObamaCare, and pushed back against Budget Committee Chairman Paul RyanPaul Davis RyanNo time for the timid: The dual threats of progressives and Trump Juan Williams: Pelosi shows her power Cheney takes shot at Trump: 'I like Republican presidents who win re-election' MORE's (R-Wis.) op-ed that suggested seeking other concessions not related to the law in order to reopen the government and increase the debt ceiling.

"Anything that comes out of this has to address the core fight, which is ObamaCare," Needham said during a breakfast hosted by the Christian Science Monitor. "The only acceptable way out of this is some sort of deal that funds the federal government without funding ObamaCare."

Needham said that discussions over tax reform and the Keystone XL pipeline were misguided, and said the suggestion by some Republicans that getting rid of the medical device tax would be enough was "laughable."

"With regards to Paul Ryan's op-ed ... this is a fight about ObamaCare," he said. "The attention of Republicans and conservatives needs to be back on ObamaCare and not on other ways out of this situation."

Needham and his organization were instrumental in pushing congressional Republicans to force a government shutdown unless Democrats agreed to defund President Obama's signature legislative achievement. Heritage Action has pressured House and Senate Republicans since August to not give in on the fight, using advertising and heavy lobbying.

The resulting government shutdown has led to a stalemate. Democrats are refusing to negotiate over GOP priorities in order to reopen the government and raise the debt ceiling, while Republicans refuse to come away empty-handed.

Polls show the GOP brand is taking more of a hit from the standoff than Democrats, though the gap isn't as wide as it was during the 1995 government shutdown.

Needham predicted Obama will "feel pain" politically for his refusal to negotiate on the government shutdown and Ryan's Wednesday morning op-ed argued that the way to resolve the impasse was to agree to "common-sense reforms of the country's entitlement programs and tax code" instead of a focus on ObamaCare.

Needham downplayed concerns that the government will default, arguing that it would be able to pay its debts even after the Oct. 17 debt ceiling date. 

He did admit, however, that such a move would roil the markets, and said short-term increase in the debt ceiling would be acceptable to the group "as long as it comes with a promise to keep up the fight" on ObamaCare using the government shutdown.

"We give the Speaker some flexibility on a short-term debt limit increase," he said. "The debt limit right now is a battle we're not engaged in."