Speaking on FS1's "First Things First," Carter said he sees Peterson's career ending with a thud in much the same manner that he experienced during his one year in Miami.
Playing in his 16th NFL season, a then-37-year-old Carter was limited to just eight receptions while playing in five games with the Dolphins in 2002.
"Did you see me in a Dolphins uniform?" the 52-year-old Carter told USA Today. "I had no business playing that season. I wouldn't have admitted it, either. These guys get so sensitive about everything. Adrian had better get over himself. We were all great. But at the end, we all smelled like a baby's diaper."
Sharpe, 50, also painted Peterson in a negative light on "Undisputed" -- much to the chagrin of the running back.
"Watching some of the things they said about me, man, it really hurt me to the core," Peterson told USA Today. "Not only are they black men, but these are people I looked up to. And these are people that made mistakes, especially Cris Carter. So some of the things that came out of his mouth, not only personally, but about me as a player -- Aw, he's washed up and this, that and the other, and he should just retire -- how dare you.
"Then Shannon Sharpe, the same thing. He said some things and I'm just like, 'Wow, I can't believe that would come out of your mouth.' I understand that people are entitled to their opinions. That's the way of the world. But they are in a position where millions of people are watching them."
Peterson wasted no time making a positive first impression with Washington.
Four days after he was signed by the Redskins, Peterson was in the lineup and rushed for 56 yards on 11 carries in Friday's preseason game against the Denver Broncos.
It was the first game action for the seven-time Pro Bowler since late November, when he suffered a neck injury and was placed on season-ending injured reserve by the Arizona Cardinals.
Washington is thin at running back after a series of injuries at that position. Rookie Derrius Guice suffered a season-ending torn ACL, and Samaje Perine and Byron Marshall each sustained ankle injuries that has sidelined them.
Peterson played in 10 games for the New Orleans Saints and Cardinals last year before injuring his neck. He rushed for 81 yards in four games, including one start, for the Saints before being acquired by the Cardinals. He added 448 yards in six games, all as a starter, for Arizona.
The 33-year-old Peterson rushed for a league-leading 1,485 yards for the Minnesota Vikings in 2015, when he was named All-Pro for a fourth time. However, he was limited to three games and 72 rushing yards in 2016 and was released by the Vikings in March 2017.
Peterson was named the league MVP in 2012, when he rushed for 2,097 yards. His 12,276 career rushing yards rank second among active players, behind only Frank Gore, and rank 12th all time.