There are certainly parallels in the efforts of the frontiersman and of those pursuing the American Dream of materialism. For, both were, as mentioned already, highly self-motivated, goal-orientated, and bold. However, the values of their quests differ tremendously. Establishing a new frontier involved the claiming of land, a tangible and worthy and fulfilling acquisition, while achieving the materialistic American Dream is empty in its hollow materialism.
If we look at Jay Gatsby as a person who has re-made himself and who has hacked out his own identity (as if out of the proverbial void/clay) then I can see a connection between Gatsby and the Frontiersman.
The frontier was romanticized for its potential, for the freedom it represented. That freedom was entirely aligned with the idea of self-determination, a trait that certainly characterizes Jay Gatsby.
In many ways, yes. For those heading West during the land and gold rush, the pursuit was land and fortune. By the time Gatsby has arrived in American history, the way to pursue these things have changed and updated. In order for Gatsby to have a chance at getting Daisy back, he has to have connections, a business, and lots of money.
As we saw from the schedule that Gatsby wrote out for himself in the last chapter, we see that he tried to cover everything that he thought would be necessary to succeed in his pursuit: sports, weight training, poise, work, studying, etc. He was trying to become everything he needed to be in order to achieve his goal.
I think that this is an interesting comparison. To a certain extent, I can buy it because both visions are seeking to appropriate the world in accordance to their own subjectivity. The frontiersman does this in subduing the wilderness, creating new domains where he is the unquestioned force of power, and never relenting in his quest for control. Gatsby is much the same in his desire to appropriate the world in accordance to his dreams and how he never shows a spirit of surrender despite overwhelming odds within such a scheme. Yet, I think that the challenge in trying to compare both is that they operate from a different starting point. The frontiersman is operating from a position of comedy, or harmony in unity. The American History where the frontier is being challenged by the frontiersman is one where he essentially wins. Few learn about frontiersmen like Daniel Boone being unsuccessful or failing in their quest. Part of the allure of American History and a part of its dialectical development lies in the success of these individuals. This, though, is not Gatsby. Jay Gatsby is different from the frontiersman because he fails where his predecessor succeeds. Gatsby's narrative is tragic, and we, as the reader, grasp this. Gatsby seeks to appropriate the world in accordance to his own subjectivity and encounters others who do the same. He loses in this quest, whereas the frontiersman, for the most part, does not encounter an adversary who is the equal to him. Gatsby's failure is both in his desire to appropriate the world in accordance to a subjectivity that does not match his external experience and in that he never recognizes the failure in doing so. In this light, Gatsby acquires a different and more tragic sensibility than the frontiersman.