During a segment on his Monday evening show, Mr Carlson showed side-by-side images of Representative Ilhan Omar and Senator Tammy Duckworth, both of whom are Democrats on Capitol Hill and were born overseas.
The chyron on screen then read: “We have to fight to preserve our nation & heritage”.
Critics said the phrase across the screen mirrored that of the white supremacist 14-word slogan: “We must secure the existence of our people and a future for white children.”
The Lincoln Project, an anti-Trump super PAC formed by Republicans, shared the image of the moment from the show and said Mr Carlson should “just put on a white hood and save us the energy”.
Other critics said Carlson had to compress the statement into 10 words instead of the 14-word phrase because of how much could be fit on the chyron.
“President Trump! You have one last chance to turn the tables, win this election and save America – and yourself ! Nominate Tucker Carlson for Vice President. This would energize your campaign beyond belief,” the white supremacist tweeted.
The Independent contacted Fox News for a comment.
— Andrew Feinberg (@AndrewFeinberg)
Just put on a white hood and save us the energy. pic.twitter.com/eSlPOqCpVh
— The Lincoln Project (@ProjectLincoln)
Tucker must be absolutely fuming that all 14 words wouldn’t fit https://t.co/Abpv6G0CWa
— Jeff Yang (@originalspin)
A feud started between Carlson and Ms Duckworth, an Iraq war veteran, after she said there should be a “national dialogue” on whether statues like those of George Washington and Thomas Jefferson should be taken down when speaking on CNN.
“You’re not supposed to criticise Tammy Duckworth in any way because she once served in the military,” Carlson said during his monologue on Monday. “Most people just ignore her. But when Duckworth does speak in public, you’re reminded what a deeply silly and unimpressive person she is.”
“It’s long been considered out of bounds to question a person’s patriotism. It’s a very strong charge, and we try not ever to make it. But in the face of all of this, the conclusion can’t be avoided. These people actually hate America. There’s no longer a question about that,” he added.
That was when the chyron displaying the controversial messaging aired.
Does @TuckerCarlson want to walk a mile in my legs and then tell me whether or not I love America?
— Tammy Duckworth (@SenDuckworth)
Attacks against Ms Duckworth from the Fox News host then heightened on Tuesday evening after he reportedly asked the senator to appear on his show.
He said a representative “informed us that before even considering our request, we must first issue a public apology for criticising” the senator.
“Keep in mind that Tammy Duckworth is not a child, at least not technically – she is a sitting United States senator, who is often described as a hero,” he added. “Yet Duckworth is too afraid to defend her own statements on a cable TV show. What a coward.”
Ms Duckworth responded to the attacks in a tweet, writing: “Does @TuckerCarlson want to walk a mile in my legs and then tell me whether or not I love America?”
The senator lost her legs in Iraq in 2004 after a Black Hawk helicopter she was co-piloting was shot down.
She is one of other female candidates up for consideration as Joe Biden‘s vice president pick.