"I'm a grown man. I've been in locker rooms," Martin told Dungy. "One incident doesn't bother me. It was the persistence.
"I felt trapped, like I didn't have a way to make it right. I thought it was best to remove myself from the situation."
Martin also claimed he was not the only victim of hazing in the Dolphins' locker room.
"There are other people that got it, too," Martin said. "I can't say why I may have gotten more."
Martin told Dungy that he wants to return to the NFL, and the former coach said he believes the 24-year-old tackle will "get an opportunity."
Martin left the Dolphins in October amid allegations he was harassed by teammates, including Incognito. New York attorney Ted Wells began an investigation ordered by the NFL in November.
"I have no problem with the normal hazing that you see in the NFL -- get a haircut, stuff like that, little pranks," Martin said. "But of a personal, attacking nature, I don't think there's any place for that."
During Tuesday's news conference to introduce new general manager Dennis Hickey, Ross was asked whether Martin or Incognito would return to the team in 2014.
"I don't believe so," he said.
He quickly tried to backtrack, but his first answer was already out.
"Well, I can't say that. Therefore, I retract that," Ross said, laughing. "One is a free agent, by the way; the other is on our roster."
Ross said he expects Wells' report to be released not long after the Super Bowl. Ross said he's communicated with the NFL and has a good idea of some of the report's contents.
The Dolphins finished 8-8 and fired general manager Jeff Ireland. Ireland signed Incognito as a free agent in 2011 and drafted Martin in the second round in 2012.
ESPN.com Dolphins reporter James Walker, ESPN.com's Ashley Fox and The Associated Press contributed to this report.