No, Gatbsy cannot erase his past. No matter how hard he works to reinvent himself, he will always be from a poor background. The American Dream is a myth. Old Money will always outweigh New Money.
Jay Gatsby has not become completely successful in erasing his past. It is obvious that there are suspicions from his party guests who hypothesize that he has killed a man, or been a spy, or was an Oxford man. He also displays a photo of Dan Cody who is a direct link to his humble past and a catalyst into his extravagant present and future.
In addition, his father shows up to the funeral after having read about Gatsby's death in the newspaper. Mr. Gatz's presence clears up quite a bit to Nick and others who were at the funeral. Mr. Gatz is in awe of Gatsby's home and possessions indicating that he is a humble man without wealth or social position. Even in death, Gatsby's past is not secure...in fact, it becomes more of an open book.
Jay Gatsby could never quite erase his humble roots. He had money, but he could never obtain the social class that comes with being born into a higher class family. Proof of Gatsby's lower social standing (lower than Daisy's social class) is found throughout the novel. The people he does business with, like Meyer Wolfshiem, are not people that people like Daisy would do business with. In ch. 5, when Gatsby awkwardly offers to repay Nick, we again see proof of this difference. In Gatsby's world, everything has a price, unlike Nick's world where people do things for people out of kindness or friendship. Later in that chapter when Gatsby shows Daisy his shirts and she weeps, the social difference is shown. He's showing off his wealth in a crass way - something those in Daisy's circle would not do. In ch. 6, when Tom, a man named Sloane, and a woman, stop at Gatsby's house while out riding, Sloane essentially snubs Gatsby. Gatsby doesn't even realize that Sloane is doing so; he does not realize that Sloane sees Gatsby as someone providing him with a place to get a cool drink and rest his horse, but not someone with whom he'd socialize. And of course, the scene in ch. 7 in the hotel in NYC, shows class difference. Tom throws the class difference in Gatsby's face during their argument. Tom knows Daisy would not leave him for someone of Gatsby's background. Gatsby does not and cannot erase his past.