In Aristotle's Poetics, the author indicates that the tragic hero is a relatively virtuous person who moves from a state of good fortune to a state of bad fortune.
The change of fortune should be not from bad to good, but, reversely, from good to bad. It should come about as the result not of vice, but of some great error or frailty... (S. H. Butcher translation)
This is certainly true of Romeo. He is no saint, but he is certainly not a depraved sinner. At the outset of the play, he is in relatively good fortune, although he is unhappy about the way his love life is going. This unhappiness, however, is nothing compared to his status at the end of the play: he has been exiled from his native land (compare Oedipus) and he thinks his wife is dead, so he kills himself.
Also, Romeo's killing himself when he thought that Juliet was dead seems like a prime example of what Aristotle meant by hamartia, which is often incorrectly translated as "tragic flaw." The Greek word harmatia actually seems to mean something more like "mistake" or "unwitting error." Thus, just as Oedipus unwittingly killed his father and married his mother, Romeo mistakenly thinks that Juliet is dead and thus kills himself.
As for how Romeo does not fit the bill of a tragic hero, this may be a bit elusive. In the tragedies that Aristotle had in mind when he wrote the Poetics, the tragic heroes would often appear to have the gods turn against them in some way (compare Apollo's prediction about Oedipus; Aphrodite's hatred of Hippolytus). The role of God or the gods in Romeo and Juliet is not as clear cut as it was in most Greek tragedies. In fact, in Romeo's case, the church, represented by Friar Laurence, actually tries to aid Romeo as much as possible. Juliet sees their marriage as blessed by God as she tells the Friar: "God join'd my heart and Romeo's, thou our hands".