I'd like to suggest that Gatsby can be both at the same time and that he may be a fool because he is so single-minded.
If Gatsby were to take a wide-view or an overview of his life after re-encountering Daisy, he could have been satisfied to live with his successes. Instead of taking a distanced approach, Gatsby goes the opposite direction and chooses to define his whole life through Daisy. If he cannot marry her, he is a failure.
This is a foolish way to think for a man of his accomplishments, even if they haven't been entirely legal.
Gatsby genuinely believes that can "repeat the past" and is absolutely incredulous that Nick would suggest otherwise. Gatsby has devoted his life, not just to recapturing Daisy's attention and affection, but creating the vision of himself he started to craft at a very young age when he wrote his personal improvement goals in the back of that book. While Gatsby may be foolish to think he actually achieve all those goals, he is not a fool, and there is a distinct difference between being a fool, and merely acting/thinking foolishly.
Gatsby is no fool. While he does make mistakes, he demonstrates integrity and the beautiful soul of the romantic who dares to pursue his dreams. Added to this, there is a genuineness about Gatsby; this is represented in the comment of Owl Eyes that Gatsby's leather bound books are real and have real pages. Because Gatsby is genuine, Nick Carraway tells him, "You're worth the whole damn bunch of them." Would a grounded Midwesterner, Nick Carraway, say this to a fool? I think not.
He's a fool. The idea that you can and should make someone fall in love with you by getting money is completely ridiculous. If someone truly loved you, they would not reject you simply because you did not have enough money. Surely he should have seen that he could never get true love from a woman who was shallow enough to withhold her love because he did not have enough money.
I think that this is a tough question for a couple of reasons. The first is that there is always a fairly close proximity between any romantic and fool. It is often difficult to find where the line is between both realities. The other reason why this is challenging is because of the characters seen in the novel, Gatsby is probably the "nicest," making it difficult for the reader to clearly draw a negative judgment. When being surrounded by a cast of characters like Tom or Jordan, one begs to find a nice person, and Gatsby is it. Gatsby is more in love with an idea, than anything else. His apotheosized vision of Daisy causes him to engage in a quest to capture her. This is not a real vision of love, but his pursuit of her is Romantic, nature. Romantics tend to pursue that which cannot be gained, seeing something that others do not see. It is foolish because a person cannot be "won," but Gatsby's ability to dream and to fit Daisy into his dreams is both Romantic and foolish. I think that Gatsby is probably a bit foolish in being able to conceive of something that is “a promise that the rock of the world was founded securely on a fairy's wing,” but this might be where he is, both at his best and worst, very human.