Rep. Bill Johnson (R-OH), a member of the House Energy and Commerce Committee set to investigate the law's implementation this week, said earlier Monday that the rollout should cost Sebelius her job.
"Absolutely. It starts at the top," Johnson said on Fox and Friends.
The White House has maintained that Sebelius has Obama's "full confidence," even as problems with HealthCare.gov, the website for online enrollment in health care exchanges, have piled up since its unveiling on Oct. 1.
President Obama defended the law as "not just a website" in a statement from the White House Monday, but conceded that the website "hasn’t worked as smoothly as it was supposed to work."
McConnell told Fox News that he "didn't think that any amount of apologizing" would fix Obamacare, but that the Republican party was best served by focusing on repealing parts of the law piecemeal until the 2016 election.
"The problem is we don't control the Senate and we don't control the White House," McConnell said. "So our ability to pull it out root and branch is not there. What is there is for Democrats to have second thoughts. Senate Democrats, many of them voted to repeal the medical device tax. As I indicated, the president decided to delay the mandate. Democrats are having second thoughts about this law as well. Since they control the Senate and control the White House, we urge them to join us and at least begin to delay parts of this because it's very, very bad for our economy."
Sebelius is "expected" to testify before the House on the matter next week, according to the House Energy and Commerce Committee.