Any action taken between players should be handled carefully. If both players are participating, or if it is a light scuffle, let it go. For example last year's "slap-gate" in Denver shouldn't result in any legal action. But if one players cold clocks another then yes, involve the police by all means. A certain amount of physical action is expected at these events, it is a full contact sport. But some things go beyond what could reasonably be expected.
But in those rare cases, it definitely should go to law enforcement.
If I go into my office and punch a coworker in the face, is it up to my business to punish me? No. The burden of punishment is on the law. Sure my business would also "punish" me probably by firing me, but that is not acceptable.
Oh, and any action towards a fan - like this ridiculous pitcher - should go to our court system. There is no reasonable expectation of going to a sporting event and being physically assaulted.