This would be easier to answer if there was a simple definition of tragedy. There are several pieces to Aristotle's definition and they are interpreted multiple ways by different critics. But, in my opinion, "All My Sons" is a tragedy, and can be seen as a classical tragedy.
“Tragedy, then, is an imitation of an action that is serious, complete, and of a certain magnitude; in language embellished with each kind of artistic ornament, the several kinds being found in separate parts of the play; in the form of action, not of narrative; with incidents arousing pity and fear, wherewith to accomplish its katharsis of such emotions. . .” (translation by S. H. Butcher)
Certainly, the play is serious and complete. It absolutely attains a fairly high magnitude. Miller's use of language is very artistic; look at the Doctor's speech about Chris having to be alone to watch his star go out. In terms of catharsis, several characters can be said to have a catharsis, but most of all, it is Joe Keller who has the catharsis. He finally comes to understand the true meaning of his actions and is undone by that. I would also say that an audience would also find the play cathartic, in that there is much to be learned about setting priorities and making ethical choices, even in the face of family need.