The outrage continues to build over a report from her department that warned of the danger of right-wing "extremists," and singled out returning war veterans as susceptible to recruitment.
Napolitano expressed regret for the reference to veterans -- but she raised eyebrows again this week when she suggested that the Sept. 11 hijackers entered the United States through Canada, even though the 9/11 Commission determined they came to the United States from overseas.
"I don't know that the secretary understands the depth of the disruption that she's caused," Rep. Michael Burgess, R-Texas, told FOX News on Thursday, referring to the report on extremist threats. "I think the appropriate thing to do is for her to step down and let's move on."
Conservatives made a stern call for her ouster Wednesday night on the House floor.
"Mr. President, fire that woman," said Rep. John Carter, R-Texas, complaining that Napolitano's comments on the controversial report were half-hearted. "To go on television and say your apology to be, 'I'm sorry you were offended by this report,' that's no apology."
Rep. Michele Bachmann, R-Minn., also said Napolitano's resignation is in order, and she should be brought before Congress for a hearing.
Napolitano on Thursday acknowledged the criticism and reiterated that the extremist report was "not well written" and should not have been released in that form. She said she would meet with the leadership of the American Legion on Friday over the reference to returning war veterans.
But she rebuffed those who say an apology is not enough.
"That's what they're going to get," Napolitano said.
She also corrected her statements on Canada, admitting that she falsely suggested Sept. 11 terrorists crossed over from Canada. "I knew the minute it came out of my mouth it was wrong," she said.
Napolitano first clarified her comments in a written statement that said: "I know that the September 11th hijackers did not come through Canada to the United States. There are other instances, however, when suspected terrorists have attempted to enter our country from Canada to the United States."
Her explanation was not so clear during the interview Monday with the Canadian Broadcasting Corporation that sparked the new controversy.
The secretary was asked to elaborate on comments about giving the Mexican and Canadian borders equal treatment, even though Canada is not experiencing a violent drug war.
"Yes, Canada is not Mexico, it doesn't have a drug war going on," she said. "Nonetheless, to the extent that terrorists have come into our country or suspected or known terrorists have entered our country across a border, it's been across the Canadian border. There are real issues there."
Napolitano was asked if she was referring to the Sept. 11 hijackers. She said: "Not just those but others as well."
This angered some Canadian officials, who called such claims an unfortunate "misconception" in media interviews.
The Canadian newspaper The National Post unleashed on Napolitano. One column called the interview a "train wreck." Another questioned how Napolitano could view the Canadian border as a "security threat on par" with Mexico's.
"Ms. Napolitano's brief interview with the CBC this week was confirmation we're dealing with an irrational senior U.S. official who can't differentiate between a secure border linking the world's largest trading partners and one that's a giant sucking sound for jobs going south and what's been described as an 'invasion' of desperate Mexicans illegally sneaking north," the column said.
Such criticism sprung in part from a speech Napolitano delivered last month at the Brookings Institution, in which she said "we shouldn't go light on one (border) and heavy on the other."
"If things are being done on the Mexican border, they should also be done on the Canadian border," she said. "This is one NAFTA, it's one area, it's one continent and there should be some parity there."
She said she intends to "visit myself" the Canadian border this spring or summer.
House Minority Leader John Boehner briefly addressed the criticism over Napolitano on Thursday.
"I think Secretary Napolitano has an awful lot of explaining to do," he said.
FOX News' Catherine Herridge contributed to this report.