The Great Gatsby is a sophisticated, serious work of fiction. It deals with realities of human existence. Fairy tales and wishes and dreams don't usually come true: in reality or in serious fiction. The characters in the novel are flawed, as are actual people, and as are character and human perceptions and judgments.
Most specifically in the novel, Jay Gatsby's idealistic view of his romantic relationship with Daisy turns out to be erroneous. It is an illusion. His view of their relationship as special and poignant, "earth shattering," if you will, is an illusion. Daisy never loved him as he loves her. The romance was based on misperception.
Gatsby, perhaps, loves as everyone should love. And he loves Daisy as everyone would wish to be loved. But love blinds him to reality (thus, the eye doctor sign). His love results only in tragedy.
Romance as a whole is contaminated in the novel: Tom and Daisy's relationship is based on money and the maintenance of the status quo; Wilson and Myrtle are miserable, disloyal (in Myrtle's case), and abusive toward each other; Jordan is amoral and Nick never commits to her or vice-versa. And Tom smacks Myrtle in the face.
If one applies romance in the novel to life as a whole, one could indeed say that romance dies in The Great Gatsby.