All he does is sit on his chair and watch TV is correct. It means he does nothing else but sit and watch TV.
I would say that sit and do are bare infinitives [without to] and in this construction an infinitive is required not a Gerund or present participle.
One more thing: in a chair or on a chair. I would probably expect in a chair here because he is watching television and he is probably sitting in an armchair. However, if, indeed, he is sitting on a wooden chair at a table, then on a chair would definitely be correct.