Seems like a distraction. I get that part of Trump’s winning strategy is that he does fight back and doesn’t let attacks go unanswered, but this one seems unrelated to the Presidency.
I think it is a distraction and that is exactly what Trump wants... He knows the media, they would rather cover this kind of nonsense than real hard news, and for that matter most of the low information electorate would rather hear this type of garbage than real news. So Trump engages in a tit-for-tat twitter with Schwarzenegger, while he is signing executive orders gutting Obama Era Dodd-Frank regulations with the other hand. And what does the media focus on all weekend? The pop culture nonsense, leaving him free to do what he wants on the political side.