It varies state to state but not much.
Some stadiums (Including Levis stadium) have a holding cell in them. Some jurisdictions even have a judge on standby for bail hearings right there. In others the stadium jail is just a holding area until transport to county is arranged. Not sure how Santa Clara county does it. In any case, He could get disorderly conduct, and trespassing charges. If he was drunk he'll get charges for that. More severe charges could come out of the investigation. By the way that was the best defense the Rams played all night. Such things are not usually handled on a national level unless there is terrorism involved or unless the crime crosses state lines or something.
Anyone in any state can file a police report at any time. The law will investigate that police report just like it would any other police report. It doesn't mean the DA will file any formal charges or that the cops will even make an arrest. It's flat out going to go nowhere. In fact I'd bet the cops just laughed their heads off when the guy left. But don't that get in the way of an anti California rant.