How do Tom and Daisy treat Gatsby in a despicable manner?
In The Great Gatsby, first of all, if by destruction you mean death, Wilson is responsible for Gatsby's death--he pulls the trigger. And Tom tells Wilson that Gatsby owns the car that hit Myrtle, so Tom plays a part as well. Furthermore, Daisy lets Gatsby take the blame and doesn't tell Tom, as far as evidence in the novel suggests, that she was actually driving the car. So she is responsible, too.
In terms of his death, Gatsby is responsible for his own death only in that he takes the blame for the accident for Daisy. His act of love and protection gets him killed, you could say.
If you're asking about more than just the death, Daisy says she loves Gatsby but then withdraws from him because, as she says, Gatsby asks too much--he demands that she announce that she always loved him and never loved Tom, and she refuses to do that.
Tom slanders Gatsby with speculation about his past, etc., in an effort to keep Daisy.
Gatsby, if you're asking about more than just the death, causes his own failure by dedicating himself to an illusion, and trying to recapture a past that never was.