The sentence is correct, but I'd interpret it to mean that he's a good author or that he's good at writing essays. I think "His handwriting is very good" is the most natural option.

We don't have a tradition of calligraphy, and since the widespread use of computers/keyboards, the average child doesn't spend much time making letter forms. So for us, 'he writes well', has no suggestion that his handwriting is good - rather that he knows how to compose a good sentence. So you would need to specify that it is his 'handwriting' that you are referring to. A more traditional term is 'penmanship', but that isn't in common usage these days.