(CNN) - Longtime Republican Sen. John McCain on Sunday described the enrollment process for Obamacare thus far as a "fiasco" and vowed to continue fighting the health care law. But he said Republicans should take a "rifle shot" approach rather than the "meat ax" strategy of defunding the law altogether, as endorsed by conservatives like Sen. Ted Cruz.
On CNN's "State of the Union," McCain was not quite ready to call on Health and Human Services Secretary Kathleen Sebelius to step down, unlike others in his party, but he supports the idea of holding congressional hearings over the issue and encouraged more efforts to dismantle the Affordable Care Act.
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"Let's find out who is responsible for this fiasco and then take the appropriate action," he told CNN Chief Political Analyst Gloria Borger when asked about Sebelius.
"But this is just the beginning of the problems," he continued. "That's why we Republicans have to keep up the fight. But we have to rifle shot it rather than go at it with a meat ax, which cannot succeed."
Nearly half a million people have filled out applications for Obamacare coverage since October 1, despite the highly publicized problems with online sign-ups, administration officials said Saturday.
"The website is unacceptable, and we are improving it," one senior administration official said. "But the underlying insurance product is good, and across the country, people are getting access to affordable care on January 1."
The Affordable Care Act is intended to serve more than 48 million Americans without health insurance. Most Americans face tax penalties if they do not have health insurance by the beginning of next year.
McCain had his own recommendation on how to fix the rocky rollout.
"Send Air Force One out to Silicon Valley, load it up with smart people, bring them back to Washington and fix this problem," he said. "It's ridiculous. And everybody knows that."
The 2008 GOP presidential nominee said he believes that the Obamacare glitches would be a bigger story if conservative Republicans hadn't staged a strategy that in part led to this month's government shutdown.
"Many ironies here, but one of them is the fiasco of this rollout has been obscured because of this … strife that's been going on in the Republican Party," he said.
Spearheaded by Cruz, House Republicans repeatedly attached anti-Obamacare provisions to must-pass spending bills, which the Democratic-controlled Senate refused to take up. The stalemate ultimately resulted in a 16-day partial government shutdown that removed $24 billion from the economy, according to an initial analysis from Standard & Poor's.
McCain has been one of many Senate Republicans to criticize the effort.
"It was a fool's errand to start with. It was never going to succeed," he said. McCain didn't single out Cruz but blamed "the whole effort" instead.
"Keep up the fight against Obamacare. But don't shut down the government and have so much collateral damage," he added.
Although Cruz hasn't ruled out trying to take the same approach when Congress faces another fiscal deadline in January, McCain said he's confident Congress will not let it happen again. "The American people will not stand for another one of these things. They just won't."
This week, the House Committee on Energy and Commerce will hold a hearing on problems with the Obamacare enrollment process. Sebelius was invited to testify. A Health and Human Services official said last week that she won't be available for the Thursday hearing, but she's in "close communication" with the committee about its request for her testimony.
Some lawmakers have called on Sebelius to be fired or resign, but Republican Sen. Marco Rubio, like McCain, didn't immediately call for her to step down over the problems. However, Rubio, along with Sen. Roy Blunt, R-Missouri, criticized her for not going to the hearing.
"Her refusal to testify and be transparent about it, I think, is undermining her credibility, and there may come a point where, in fact, she will have to resign, largely because she no longer has the credibility to do the job," Rubio said on "Fox News Sunday."
On the same program, Blunt argued that there will come a point when "she'll have to testify."
"I don't think she can refuse to answer questions about this," he said.
– CNN's Kevin Bohn and Greg Clary contributed to this report.
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