Uh, if it goes that far

There's less than 1% chance it would go that far. The contract between Twitch & Doc is a private business agreement. Aside from possible tax division stipulations, the 'law' / government does not care about this agreement. If Doc breaches contract, it is entirely up to Twitch legal to both pursue litigation and prove to a judge that they are due compensation for Doc's actions. The government is not involved in any way until the judge reaches a decision.

Now, if the judge issues a court order for Doc to pay Twitch restitution for his actions, then......he does it. He's a wealthy guy with several high-profile brand deals in the mix, so he likely has a massive incentive to simply payout Twitch, regardless of cost. He may even have a new streaming platform lined up already, and the monetary value of switching to another platform (or even starting his own) would likely be far more lucrative than just stagnating within the Twitch ecosystem. Hell, another streaming company may have even offered to pay out his contract breach if required.

Even if Doc just straight-up told the judge "I'm not paying them/I'm not following the court order," they would just get the Sheriff's department to seize enough assets to equal the restitution. These seizures are more polite than you would think, so unless Doc decided to just start throwing punches, deny the Sheriff entrance, or attempt to defraud the court auditor (about the value of his assets), he's not going to jail.

The other possibly is a judge that just goes "lol costumes and videogames" and throws out the suit.