Is Edward II a genuine tragedy?

It depends entirely how you define tragedy.

In the Elizabethan sense, you'll notice that criticism as a whole doesn't distinguish very clearly between "histories" and "tragedies". Plays, like "Edward II" which deal with real historical characters - although, in very definitely with a big pinch of dramatic licence - are usually called "histories", although they can also be considered "tragedies". So in the generic, dramatic sense, "Edward II" is both a history and a tragedy: a tragedy which deals with historical characters.

More usually, though, critics write about tragedy as a literary genre in itself, and one which adheres to specific rules (these rules usually taken from, or at least, inspired by, Aristotle's Poetics). In very, very general terms, Aristotle's idea of a tragedy is the fall of a noble person from high prosperity into misery (or usually death) - and as the result of a mistake (or "hamartia": absolutely not to be translated as tragic flaw - a concept nothing to do with Aristotle) that he or she makes.

Edward does fall from the highest office in the land - the crown - to a miserable death, imprisoned. The real question for interpreters of the play is, if the play is a tragedy, why? Is it his weak rule, his love (in whatever sense for Gaveston), or something else?