Okay, well let's see if we can sort this out. First, Aristotle, a Greek, was basically the first person to clarify what makes a "true" tragic character. He believed (1) the character must of noble stature, such as a person of royalty-think of Shakespeare's King Lear or Hamlet, Prince of Denmark, for example (2) the character must have some tragic flaw, such as excessive pride, impulsiveness, rejecting the Gods, etc. (3) the flaw must somehow lead to his death, and (4) he must have a catharsis, or a cleansing of emotions, before he dies. This is kind of like a lesson to the audience, like a "Don't make a mistake like I did" kind of message.
A modern tragedy, though, might not have the "noble" character that a true Greek tragic character would have; for example, Arthur Miller's Death of a Salesman would not be considered a true Greek tragedy because (1) he's only a salesman, and not a very good one at that, and (2) even though he dies at the end, he really doesn't ever truly confess and regret his faults before he dies.
I suppose there are some elements of a modern tragedy in A View from the Bridge, such as Eddie's tragic flaw as his obsession with Catherine. And in the end, he dies in a scuffle with Marco. I would categorize this play as a modern tragedy. It has some elements of a Greek tragedy, but not all of them.