You should get out of that relationship immediately.
Apart from the fact that he's clearly paranoid and and mentally unstable, he's crossed the line that I think most people would consider a deal breaker when it comes to relationships: Physical violence.
It sounds to me that he's either very insecure. To accuse a partner of cheating like that, and then subject them to violence is a common sign of this. Unfortunately, rather than dealing with his own inadequacies, he's taking out his problems on you.
And, no, you won't be able to fix him - you're not a psychiatrist or trained mental health professional (unless you are, in which case how did you not spot this personality disorder before things got serious?!).
But seriously, you are actually hindering his acceptance/recovery of his problem by staying with him, as he will always have you to blame and vent his anger on this way.
You are also putting yourself in serious danger. If he's willing to assault the person he's meant to love and protect, how far will he go when he's angry/drunk/high/feeling depressed. It just takes one bad day, a burst of rage, and you could end up seriously injured or worse.
Unfortunately, I have a couple of friends who have been in similar situations, and they really struggled to call time on things, and even returned to their boyfriends' after a while.
I think a lot of people underestimate how difficult it is to walk away from an abusive relationship. It's a complex situation. But you have to be stubborn with yourself and remind yourself of the consequences of people who don't act on the warning signs.
