"I would do anything and I will continue to do anything I can to stop the train wreck that is Obamacare," Cruz told ABC's "This Week," when asked whether he would rule out pushing the government to the brink of another shutdown in January unless the Affordable Care Act is defunded.
"Have a little bit of self-restraint," former Florida Gov. Jeb Bush said on the same program in a message to Cruz.
"It was a mistake to focus on something that couldn't be achieved," said Bush, who, like Cruz, is mentioned by some political analysts as a possible 2016 GOP presidential hopeful.
"Shutting down the government, in my view, is not conservative policy," Senate Minority Leader Mitch McConnell, R-Ky., said on CBS's "Face the Nation." "There will not be another government shutdown. You can count on that."
"It will not happen," Sen. John McCain, R-Ariz., told CNN's "State of the Union." "The American people will not stand for another one of these things."
"Let's face it -- it was not a good maneuver," Sen. Orrin Hatch, R-Utah, a supporter of the deal that ended the showdown, told The New York Times. "And that's when you've got to have the adults running the thing."
"The tactic of defunding the government unless [President Barack Obama] repealed his signature issue was as poorly designed as Obamacare itself, almost," Sen. Lindsey Graham, R-S.C., told "Face the Nation."
Obama was expected Monday to declare the computer troubles plaguing his flagship healthcare reform unacceptable as his administration launches an urgent "tech surge" to fix them.
Those troubles include computer crashes, slow responses and other glitches.
Cruz told "State of the Union" he was not bothered, even "remotely," by the growing criticism by senators of his own party.
"I don't work for the party bosses in Washington. I work for the people of Texas, and I fight for them," he said. "The reason people are frustrated all over country is that far too many people get elected and they think they're there to be part of the club."
He criticized fellow Republican senators for folding in the shutdown fight with Obama.
"I think Senate Republicans should have united," he told the CNN program. "Senate Republicans should have united and supported House Republicans. The one hypothetical that I really think is worth thinking about is how would this have played differently if, when the House stood up and led, Senate Republicans had marched into battle side by side and said we are united and saying we should fund government but we should not fund Obamacare."
Cruz joined several other conservative lawmakers in calling for the resignation of Health and Human Services Secretary Kathleen Sebelius, whose department oversees the online health insurance exchanges, in which technical problems have hampered enrollment.
McCain didn't say Sebelius should resign but called for congressional hearings to see who's responsible for the glitches.
"Let's find out who is responsible for this fiasco and then take the appropriate action," he said. "But look, this is just the beginning of the problems associated with a massive restructuring of one fifth of our economy. And there's going to be a whole lot more problems associated with this before it's done."
The House Energy and Commerce Committee has asked Sebelius to testify at a hearing this week. She has declined an invitation to attend, a response committee chair Fred Upton, R-Mich., called "wholly unacceptable."
Sen. Dick Durbin of Illinois, the chamber's No. 2 Democrat, told "Fox News Sunday" Sebelius will testify.
"Ultimately, Secretary Sebelius will testify before Congress -- you know that," Durbin said.