No he won't. He won't miss a single game/quarter, well outside that he's a 3rd string RB. What he did was stupid, but it's not like it was that big of a deal. It's not a drug arrest or an assault.
The only thing that makes it significant is he has had troubles before with team rules and attitude. He's been suspended for games and stuff a few times in the past. That history would lead more towards being kicked off the team if anything. Hasn't happened yet, so probably won't.