"I appreciate that. I appreciate that," Carlson replies.
In a comment posted alongside the video, Bailey criticises Carlson over what he called "vaccine misinformation".
He wrote: "It's not every day you get to tell someone they are the worst person in the world and really mean it! What an asshole!
"This man has killed more people with vaccine misinformation, he has supported extreme racism ... and does more to rip this country apart than anyone that calls themselves an American."
Last week, as some Fox hosts asked viewers to get vaccinated, Carlson continued to urge viewers not to "comply mindlessly" with public health guidance.
In a statement, Fox News said: "Ambushing Tucker Carlson while he is in a store with his family is totally inexcusable – no public figure should be accosted regardless of their political persuasion or beliefs simply due to the intolerance of another point of view."
The fishing store, which coincidentally shares the same name as the man in the video, posted a statement to distance themselves from Bailey's actions.
"On July 23rd, a well-known television personality, Tucker Carlson, was affronted while shopping at Dan Bailey's Outdoor Company," the statement read.
"Coincidentally, the person engaging Mr Carlson was a local resident named Dan Bailey. This person has no affiliation with our business, other than he shares the same name as our founder, who passed away in 1982.
"To be clear, we treat every customer equally and respectfully. Our staff was professional and cordial to Mr Carlson, as we are with all of our customers."
The clip has been shared multiple times on Twitter, including by Donald Trump Jr, who wrote: "Is the loser who went out of his way to have someone video him harassing Tucker in public for some viral content the model for the new Pregnant Male Emoji? The likeness is uncanny! #SoBrave."
Rightwing author Dinesh D'Souza said of the footage: "This is a shocking video. This man should be arrested for what he did to Tucker Carlson in front of his daughter."
Carlson has not yet publicly responded to Bailey's video.