If one is to look carefully at arguably the most celebrated tragic hero, Oedipus, and compare them to Gatsby, there are a number of similarities that could certainly be used to suggest that Gatsby is in fact a tragic hero.
Oedipus' most obvious flaw was his inability to see the forest for the trees, so to speak. He was very careful and logical in his ability to analyze a situation and to plan to rid Thebes of all the problems he was tasked with solving but he was unable to see that he was responsible for many of them. This contradiction is what leads the audience to sympathize with him and create the feeling of a tragic hero.
Gatsby, desperate to rekindle his love affair with Daisy is driven and incredibly successful in making himself into the type of man he thinks she needs in order to be happy. He is so focused on this that he cannot see that in some ways Daisy has already moved on, that she was never as into the whole thing as he was and that she will never accept the complications of life with him when she has what she basically needed with Tom.
This inability to see what is right in front of them is certainly a tragic flaw and in this case one that suggests that Gatsby is certainly tragic in the way a tragic hero is designed to be.