If the subject is a baby, 'fretting' would work on its own. Otherwise, as Alasdair says, you'd need to say why he was fretting.

Yes, this is correct, but does not necessarily sound like a natural sentence. I would say that you need to finish the idea off with some more detailed information : what is he fretting about? It might look like this: "He is fretting about ..." or "He is fretting because..." Hope this helps. Alasdair