Babineaux said he is embarrassed and included Falcons owner Arthur Blank, general manager Thomas Dimitroff and coach Mike Smith in his apology.
"First and foremost, I want to apologize for being a distraction to my family, to my teammates," Babineaux said in his first public statement since his arrest Dec. 10.
"I want to apologize to Mr. Blank, coach Smith and Thomas Dimitroff and also the fans, most importantly. I'm embarrassed to be standing here in front of you guys, but as you guys are all aware, it's an ongoing legal matter and I can't talk about it or answer any questions. Thank you."
The apology came one day after the release of a Gwinnett County police report that stated Babineaux told officers he was smoking marijuana while driving with a friend. Police said they found about 1½ ounces of the drug in Babineaux's car.
According to league rules, NFL commissioner Roger Goodell will decide whether Babineaux will enter the league's intervention program and face a four-game suspension because of the arrest.
NFL spokesman Greg Aiello said Tuesday the league will wait for the case to move through the legal process.
Babineaux has started every game and has 45 tackles. He leads the team with eight tackles for losses and shares the team lead with five sacks.