Sen. Ted Cruz (R-Tex.) said Sunday that lawmakers should address the debate over raising the nation's debt ceiling only after the standoff over the government shutdown is resolved.

"I want to worry about the debt ceiling after we get through the CR," Cruz said on CNN's "State of the Union."

Cruz said his uncompromising posture on shredding the Affordable Care Act in the short-term budget debate has "not remotely" hurt the Republican Party brand. But some Republicans, like Rep. Peter King (R-N.Y.), have criticized Cruz's strategy.

"This was a strategy doomed to failure," King said on "Fox News Sunday."

Cruz said Republicans should look for three things in a debt ceiling deal, including a way to go after Obama's health-care law.

"We should look for some significant structural plan to reduce government spending. Number two, we should avoid new taxes. And number three we should look for ways to mitigate the harms from Obamacare," he said.