Winston was hit with his suspension last month after a long-running investigation into his groping of the female driver in Arizona two years ago.
He initially denied the allegations but apologised after the National Football League imposed a three-game ban.
Winston will miss the Bucs' first three games of the upcoming NFL campaign -- September 9 at New Orleans, September 16 at home against reigning Super Bowl champion Philadelphia and September 24 against visiting Pittsburgh.
Winston told reporters Thursday he had addressed team-mates over his suspension.
"I talked to my team-mates yesterday and I apologised to them, because I don't want to be a distraction. Just want to focus on this season," Winston said.
The 24-year-old said he had "hard work to do to regain the trust" of those who had supported him.
He also vowed to be a role model for his newborn son and fiancee.
"I'm gonna have to teach that man to respect women and live his day-to-day life [the right way]," Winston said.
"I've really (taken) it upon myself to be proactive and make sure I'm being a great example for him and my fiancee."
Head coach Dirk Koetter was critical of Winston's conduct.
"It's disappointing that Jameis put himself in that position and put our team in that position," Koetter said.
Back-up quarterback Ryan Fitzpatrick will start the first three games in Winston's place.