The term "romantic" has several definitions, including...

...fanciful; impractical; unrealistic...imbued with or 

dominated by idealism, a desire for adventure, chivalry...

It is safe to say that Jay Gatsby in F. Scott Fitzgerald's The Great Gatsbyis a romantic. At the very beginning of the story, in Chapter One, Nick Carraway (the narrator) describes Jay as a man of "romantic readiness:"  

If personality is an unbroken series of successful gestures, then there was something gorgeous about him, some heightened sensitivity to the promises of life.… [Gatsby had] an extraordinary gift for hope, a romantic readiness such as I have never found in any other person and which it is not likely I shall ever find again.

One example can be found in Gatsby's idealism about his own beginnings. In Chapter IV, Gatsby tells Nick that he inherited a great deal of money from his family before the war. However, in Chapter Six, the truth about Jay and his past is exposed. His real name is James Gatz, and he has invented whohe wants to be. With this new sense of self in mind, he never looks back.

I suppose he'd had the name ready for a long time, even then. His parents were shiftless and unsuccessful farm people—his imagination had never really accepted them as his parents at all. The truth was that Jay Gatsby of West Egg, Long Island, sprang from his Platonic conception of himself. He was a son of God—a phrase which, if it means anything, means just that—and he must be about His Father's business, the service of a vast, vulgar, and meretricious beauty. So he invented just the sort of Jay Gatsby that a seventeen-year-old boy would be likely to invent, and to this conception he was faithful to the end.

There is a fanciful quality about how Jay's heart operated..."in a constant and turbulent riot."

Each night he added to the pattern of his fancies until drowsiness closed down upon some vivid scene with an oblivious embrace. For a while these reveries provided an outlet for his reality, a promise that the rock of the world was founded securely on a fairy's wing.

Jay Gatsby also falls deeply and irrevocably in love with Daisy. Meeting again years later, she is married to Tom Buchanan, but Jay loves her no less. eNotes' Character Analysis of Jay Gatsby notes...

Perhaps attracted to her material value, she becomes his sole reason for being. 

This is hardly the behavior of a man steeped in realism, but one searching for an element of perfection that does not exist: Daisy is not perfect, and she is not his.   In what is described as "romantic idealism," in Chapter Six, Gatsby is sure that the world can be manipulated to suit his desires:

“You can’t repeat the past.”   “Can’t repeat the past?” he cried incredulously. “Why of course you can!”

To do so is impossible, but Gatsby's romantic streak will not consider it to be an impossibility at all.