Why is Gatsby so shy?

There is, of course, an interesting and amusing contrast between Gatsby's identity as a hard-boiled Prohibition gangster and his shy, awkward behavior when he first meets Daisy at Nick Carraway's cottage. He seems almost as bashful as a high-school boy on his first date. For example:
He followed me wildly into the kitchen, closed the door, and whispered: "Oh, God!" in a miserable way.
"What's the matter?"
"This is a terrible mistake," he said, shaking his head from side to side, "a terrible, terrible mistake."
"You're just embarrassed, that's all."
Gatsby, who typically seems so poised and confident at his parties and while overseeing his criminal empire, remains embarrassed even when Daisy and Nick come over to his mansion and he shows them around. Perhaps the reason F. Scott Fitzgerald puts so much emphasis on Gatsby's discomfiture now that his dream has been realized is that it explains why Gatsby has taken such a long, roundabout way of fulfilling it.
Nick is the narrator. Everything has to happen through his point of view. Nick is only drawn into this love affair because Gatsby has been trying unsuccessfully to lure Daisy to one of his lavish parties, because Nick just happens to be Gatsby's neighbor, and because Gatsby learns that Nick is Daisy's cousin.
It is Gatsby's shyness that has taken him such a long time to get around to meeting Daisy after all these years. Another man might have just called her on the telephone and asked her to meet him for tea. But then Nick would not have had such an important role in their reunion, and he would have known nothing about what transpired when the two former lovers met again. Gatsby's shyness with regard to this girl of his dreams is the explanation of the big mansion, all those wild, expensive parties, and his friendship with Nick. It explains those beautiful shirts, the big yellow roadster, and virtually everything else.
But most of all it was necessary to explain Nick's involvement in Gatsby's and Daisy's love affair. Nick not only presents Gatsby with the opportunity to meet Daisy, but without Nick it seems likely that Gatsby would have muffed the whole reunion and never gotten Daisy to his house at all. Nick gives him the encouragement he needs and also makes it more "proper" for Gatsby to invite both of them to his house rather than summoning the necessary courage to invite Daisy to come alone. After all, Gatsby had a great deal more in mind than just "meeting" Daisy. He wanted to have a romantic long-term liaison with her and ultimately persuade her to leave her husband and marry him.
This sort of shyness is probably familiar to all of us. It is a panic that often accompanies true love. Love makes us all feel foolish. The panic can cost us golden opportunities which we may lament for the rest of our lives.

However, there could be a less innocent reason that Fitzgerald makes Gatsby so shy. Maybe he is not nervous, but trying to make Nick feel at ease about encouraging his romantic pursuit of Nick’s married cousin.
Nick is supposed to be a clean-cut young man from a respectable Midwest background. Yet he allows himself to be put into the position of a panderer for a gangster who wants to commit adultery with Nick's cousin, Daisy, and then steal her away from her husband.
Even when Nick leaves Daisy alone with Gatsby, the normally poised and sophisticated gangster is still suffering from love-sickness and panic. This softens the morally questionable nature of what is actually happening. Regardless of Gatsby's shaky nerves, he has managed to get Nick to lure Daisy to his cottage for a surprise meeting, then to get Nick and Daisy to accompany him to his mansion, and finally to get rid of Nick so that (as we all know without being told) he can take Daisy to bed.
Nick is the narrator and therefore always in the spotlight. The others can come and go, but Nick is always on stage, so to speak. He keeps bragging about his tolerance and broad-mindedness, but he is letting himself be used by a man who specializes in using others for his own ends; and the only way Fitzgerald could soften the damage to Nick as a character was to make Gatsby suddenly panic.
When Nick says goodbye to them at Gatsby's mansion, Daisy seems to have taken charge of the situation. Nick's shady role in this reunion has been smoothed over.
They had forgotten me, but Daisy glanced up and held out her hand. Gatsby didn't know me now at all. I looked once more at them and they looked back at me, remotely, possessed by intense life. Then I went out of the room and down the marble steps into the rain, leaving them there together.
Nick is in the clear. He has actually been "forgotten." It is as if he had no part in this affair. He is not responsible for what has happened or what will happen. It is as if it all happened by itself. And he just happened to be there.