How is Gatsby an idealist?

Gatsby is an idealist because, based on a brief romance with Daisy Buchanan, he builds an ideal picture of her as the only woman in the world for him. He dreams of her and longs for her as he stands on the grounds of his estate, looking across the bay to the green light at the end of her pier. He builds his life around reconnecting with her and starting over with her as if the five intervening years have not occurred.
What Nick loves about Gatsby is the audacity and grandeur of his dreaming. Nick likens it to the ideals of the earliest sailors seeing the "green breast" of a new world and dreaming about setting back the clock, starting everything over in this new land—and this time getting it right.
Gatsby doesn't just dream of his ideal; he goes after it. He thinks it is possible to create the life he wants. That is what an idealist does, and if an idealist is doomed to failure, it is, after all, the dream that counts.
Gatsby's idealism elevates him to greatness in Nick's mind, infusing him with a heightened sensitivity to the possibilities of life.

Jay Gatsby is the quintessential idealist who desperately holds onto the delusional belief that Daisy Buchanan will leave her husband, daughter, and life of luxury behind to marry him. Jay Gatsby's genuine belief that he could marry a woman of Daisy's social status and caliber illustrates his idealistic tendencies. Gatsby unknowingly objectifies Daisy and associates her with wealth, prestige, and the American Dream. Gatsby's idealism prevents him from accurately perceiving Daisy as a flawed human being who would never jeopardize her secure life of luxury to be married to a notorious bootlegger.
Gatsby's idealistic personality is also revealed in his belief that he can recreate the past. He fails to consider Daisy's romance with Tom and completely dismisses the fact that she has a daughter, which is evidence of her meaningful relationship with him. Tragically, Jay Gatsby continues to pursue Daisy. What makes Jay Gatsby the prototypical idealist is his refusal to accept the reality of Daisy's situation and realize that he cannot recreate the past. Gatsby is hopelessly optimistic and will not allow himself to give up on attaining the woman of his dreams, even after it is evident that she has no interest in leaving Tom for him.

The perils of idealism manifest symbolically in this novel. Jay Gatsby’s character is the essence of idealism. Despite everything, he still yearns for a dream life with a dream woman. He puts Daisy Buchanan on a pedestal, and as a result, loses his ability to see with clarity. Gatsby’s “love” for Daisy is not necessarily true love; it is infatuation based on objectification. She becomes the symbol of everything he wants, rather than the complex human being she really is. Because of his tendency to romanticize, Gatsby cannot see Daisy’s selfishness. He puts on rose-colored glasses and cannot see clearly.