I am not sure whether to interpret this question as romantic in terms of women or in terms of the American dream. Gatsby did accomplish the American Dream, but it did not make him happy. He did not really succeed in getting the girl he wanted either.
Jay Gatsby is dedicated to a dream, to a sense of his own potential, and to a need for self-improvement. These fixations all can be described as romantic. And we can also say that Gatsby succeeds in his pursuits, for a while at least.
The question of heroicism in regards to Gatsby really is answered by the prominent role he plays in this very famous novel. He is not a traditional hero, or even a good person, but he is the central figure in a tale of compromised and murky morals and he stands out a someone with a pure, shining dream that is unsullied to the end.
Jay Gatsby is a romantic hero because he sacrificed his life for the woman he loved. Jay's life, after meeting Daisy, centered around one goal: to make enough money to get Daisy back. He dreamed of recapturing what he and Daisy had before he left for the war five years prior to the start of the story. He bought his house to be close to Daisy. He held parties hoping Daisy would come to one of them. He asked Nick to arrange the tea party so he could see Daisy again. When Jay, Nick, Daisy, Tom, and Jordan are in the hotel in New York City, Jay tells Tom that Daisy is going to leave Tom to be with him. All the characters know that isn't going to happen except for Jay - he believes Daisy will leave Tom. When Daisy hits and kills Myrtle Wilson, Jay is ready to take the blame. Nick suggests to Jay that he leave town, just to be safe, but Jay won't leave Daisy. He believes that Daisy needs him, but she leaves town and then doesn't even come to his funeral. Everything Jay did, he did for Daisy. That makes him a romantic hero.