Why doesn't the deputy get arrested and charged after clearly assaulting the billboard manager?

I wanted to ask the exact same question when I watched it.

I think you must take a few things into account:

• The "beloved" chief Willoughby has just died. Everyone is still sad/mad about that, as shown right before that scene (everybody is crying, one officer broke a chair, Dixon fainted, ...). No one is in the mood of arresting someone for beating (not killing) another, specially if that someone is a fellow officer and friend.
• Everyone thinks that the billboards are the reason chief Willoughby killed himself. Even Willoughby himself predicted that in the letter he wrote to Mildred. So Dixon whooping the ass of the guy who agreed on putting up those billboards is well-liked by the other officers (who didn't even try to stop him or check the manager after he has just been thrown out of the window).
• It is mentionned throughout the movie that Dixon has tortured a black man while interrogating him (or what he likes to call "persons of color torturing") and yet chief Willoughby kept him in hopes of becoming a better cop someday (as explained in the letter). So, the other officers (who deeply love and respect the chief) won't arrest Dixon for "beating up a guy" on the same day the chief died, knowing that the beating actually happened because of the said death itself.

But he was kicked out of the police force by the new chief.

I do agree with you that he should be arrested and put behind bars, but I think what came into play is:

• There is always a grey area in how rules (including laws) are carried out.

Come into my head, there are a few examples:

• In Training Day, Alonzo follows and stops a whole bunch of university students who have just purchased drugs. Instead of arresting them, Alonzo intimidates them and gets all of their purchased drugs from them. Of course, you can argue that Alonzo himself is a corrupted cop, but his protege, who is on scene with Alonzo, does not protest against Alonzo's decision.
• In Training Day, Alonzo beats up two would-be rapists instead of taking them into custody, despite of his protege protests against his decision.
• In Collateral, two police officers stop Jamie Foxx's taxi and ask him to pop his trunk but the two officers let him go even they receive a 911 call about shooting nearby.

Back to officer played by Sam Rockwell, I would imagine in a small town, arresting an officer has never been a tradition. Rules are not exactly carried out to the letters, another example is:

• After losing his job as a police officer, Sam can still get back into the police station and even access case files.

In a just world, he would have been arrested, but the film is definitely tapping into 2017 ideas of police accountability where punishing police for anything is extremely rare. The fact that he was even fired at all was probably only made believable by the fact that it was witnessed by a new, out-of-town, black police chief. In a town where he would have had the sympathy of most of his white peers for what he did (given the popularity of Chief Willoughby and the the fact that most of the town thought his suicide was because of the billboards put up by the victim), it would have been very difficult to convict him anyway.