The segment got attention days before, after one of the Redskins fans told The Washington Post she left the confrontation in tears and felt so threatened she called police. One of the American Indians said the group accused the fans of supporting a racist mascot.
The "Daily Show" segment initially shows the two groups being interviewed separately. At the end, the American Indians walk in on the Redskins fans.
Before airing the segment, Stewart told viewers that his program doesn't broadcast pieces if a subject has been intentionally misled or their comments misrepresented.
"We generally don't want people to participate in the show to have a bad experience," Stewart said. "We work very hard to find real people who have real beliefs and want to express those beliefs on television, and we work hard to make sure that the gist of those beliefs are represented accurately, albeit sometimes comedically, on our program."
The segment showed very little of the confrontation between the two sides. Before they were in the same room, the group of American Indians said that the Oxford dictionary defines the word "redskin" as being "dated or offensive" and that traditionally, the word means "proof of Indian kill."
One of the Redskins fans said changing the team name would be "like losing a family member."
Virginia state Sen. Chap Petersen, a Democrat who represents Fairfax, recently helped create the "Redskins Pride Caucus" to show support for the beleaguered football team's name.
He said Friday that he is representing the four Virginia residents who appeared on the show. He said that he sent a letter to the show revoking their consent to appear on the show and that it was obtained under "false pretenses."
Petersen said his clients have not decided whether they will take legal action against the show.
Redskins owner Dan Snyder is shown in the segment saying previously that the team name represents pride, honour and respect and that those who are offended by it are taking it out of context.
The segment ends with an expletive and a message to Snyder that he's on the wrong side of history and should change the team's name.
It's the second time this week that the debate over the Redskins name made national television. The series "South Park" dedicated its entire season premiere Wednesday to skewering the name.