In another segment, Carlson is heard calling Iraq a "crappy place filled with a bunch of, you know, semi-literate primitive monkeys" and jokingly using a homophobic slur.
After the recordings were published, some companies decided to pull their advertisements from Carlson's show.
Debby Jennings, a spokeswoman for SHEEX, a bedding company, told the Washington Post on Wednesday that "due to the inappropriate statements of Tucker Carlson that have recently come to light, SHEEX has made the decision to cease advertising on his television programme, Tucker Carlson Tonight."
Pharmaceutical company Astra Zeneca tweeted Monday that it had ceased advertising on Carlson's show and would not resume in the future.
In a statement to the Washington Post, Carlson was defiant. "Media Matters caught me saying something naughty on a radio show more than a decade ago," he said. "Rather than express the usual ritual contrition, how about this: I'm on television every weeknight live for an hour. If you want to know what I think, you can watch. Anyone who disagrees with my views is welcome to come on and explain why."
Fox News Channel spokeswoman Carly Shanahan responded to a request for comment by resending a December statement on boycott efforts against Carlson's show.
"We cannot and will not allow voices like Tucker Carlson to be censored by agenda-driven intimidation efforts from the likes of Moveon.org, Media Matters and Sleeping Giants," the statement said.
Pirro also faced wide condemnation — including a rebuke from Fox News Channel — for suggesting on her Sunday show that Omar, who has criticised pro-Israel lobbyists, did not support the Constitution because she was Muslim and wore a hijab.
"Omar wears a hijab, which, according to the Koran 33:59, tells women to cover so they won't get molested," Pirro said on Sunday. "Is her adherence to this Islamic doctrine indicative of her adherence to sharia law, which in itself is antithetical to the United States Constitution?"
Novo Nordisk, another pharmaceutical company, told the Post that it was "re-evaluating" its ads on Pirro's show. Spokesman Ken Inchausti confirmed that Novo Nordisk will continue to advertise elsewhere on Fox News.
"We absolutely condemn Ms Pirro's comments, which are offensive and completely contrary to our values," a spokesperson for Letgo, a website for buying and selling used goods, told the Hollywood Reporter.
NerdWallet also told the Hollywood Reporter that "we're no longer advertising on this show and don't have plans to in the future".
ThinkProgress reported that Botox manufacturer Allergan and GreatCall, a health technology company, would not advertise their products on Pirro's show going forward.
Pirro later denied saying Omar was "un-American". "My intention was to ask a question and start a debate, but of course because one is Muslim does not mean you don't support the Constitution," she said.
Fox News Channel's hosts have previously faced companies pulling advertisements from their shows over controversial comments.
Companies withdrew their ads from Laura Ingraham's show last year after she mocked one of the young Parkland gun control advocates for being rejected from colleges. In December, more than a dozen companies pulled ads from Tucker Carlson Tonight after Carlson implied immigration "makes our own country poorer and dirtier and more divided". These advertiser decisions, however, don't tend to affect the company's bottom line and the commercials can be aired at other times.
