Before Aristotle's Poetics, catharsis was strictly a medical term meaning evacuation, purification, or cleansing. Of course, Aristotle's use is metaphoric: "It is the human soul that is purged of its excessive passions." So, in concurrence with the previous posts, there is, indeed, no catharsis in F. Scott Fitzgerald's The Great Gatsby since there has been no passion aroused in the reader by his narrative, as well as the other reasons already stated.
From beginning to end, the reader of The Great Gatsby is presented a tableau of the Jazz Age and its participants; disillusion prevails in this tableau, there is no nobility of character, no "act of injustice" that Gatsby commits. He is tragic in the sense that he attempts to achieve greatness, and when he fails, he meets his fate with courage and nobility of spirit. The reader senses this when he unfailingly supports Daisy, standing for long hours outside her window, refusing to relinquish his dream. And, although the reader is saddened by his lonely defeat, there is no high passion involved in his tragic ending.
I am in total agreement with the previous post. When I read the question, my first reaction was that, "Wait, is there one?" I think that the presence or overarching notion of a catharsis undermines what Fitzgerald sees as the fundamental problem with the "flapper" era. This social order is shallow, lacking the emotional quotient, or refusing to acknowledge, to see the complexity of human beings. Catharsis would presuppose a level of depth that the Toms, Jordans, and Daisys might not be willing nor able to accept. If there is any notion of it, Nick's departure from the social scene might be it. In constructing a novel in the modern sense of tragedy, whereby an individual experiences the implicit destruction in appropriating the world in accordance to their own subjectivity, Fitzgerald does not capitulate to something that brings unity and symmetry to this vision. Rather, his construction is of a world where Catharsis is sorely needed, but completely lacking.
Concerning The Great Gatsby, I may be in the minority, but I don't think catharsis is present in the novel. It is considered a modern tragedy, but I don't think a modern tragedy necessarily involves catharsis.
The end is not neatly tied up in the novel. Gatsby's death is an injustice. Deaths in traditional tragedies may be unjust, too, such as in Hamlet. But in Hamlet the evil is purged. No such purging occurs in Gatsby. Tom is alive and well and as self-satisfied as ever. Daisy escapes without consequence. Jordan, egocentric as she is, has suffered no loss in status, etc. The world is not better off because Myrtle, Wilson, and Gatsby are dead (see the last few pages of the novel).
If the conclusion is circular or tied up or neat, in any way, it's just that Nick will continue to dream as Gatsby dreamed, and, who knows, maybe some day these dreams will come true. But the state of affairs is as it was before. Nothing significant has changed, except that a man who loved as few others have loved has failed, and been killed in the process. This is not catharsis.