The answer here is actually ridiculously complicated.

The long and short of it is that by stepping into the ring they agree to some element of violence beyond what is normally acceptable in society, and to an extent that even goes beyond stuff that happens within the rules of the game.

It might seem like it's straightforward - if they're fighting or otherwise being violent within the context of the game, it's fine. But the reality is way blurrier than that.

Take a look at hockey, for instance, where fighting is definitely against the rules but fighting (and getting penalized for it) is an unwritten but expected part of the game. Yet at the same time there are clearly hockey fights that are criminal. Where do you draw the line?

There's a legal doctrine surrounding violence that is "incidental to the sport" - not explicitly consented to or part of the game, but that nonetheless can be considered part of the environment. For instance, in a relevant decision related to hockey (unsurprisingly, hockey is where most of the case law on the subject comes from lol) a judge stated:

given the permissiveness of the game and the risks that the players willingly undertake, I find it difficult to envision a circumstance where an offense of common assault as opposed to assault causing actual bod-ily harm could readily stand on facts produced from the incidents occurring in the course of a hockey game played at [the NHL] level

To put it more bluntly - the judge is actually skeptical that you even can commit simple assault in an NHL game, because for an incident to be violent and egregious enough to go beyond what is incidental to the sport, it would also necessarily rise to the level of aggravated assault. I think something similar could apply here, and the fighter is quite lucky he didn't actually land any nasty blows on the ref.

I personally doubt that he will be successfully prosecuted for this. But the thing is that these cases just don't come up very often in the first place, because people rarely want to prosecute, so it's a real grey area and it could go either way.