Gatsby's conflict with his self certainly relates to his conflict with Daisy. His problem with self is an issue of identity. Having a specific identity would have earned him acceptance with Daisy. As it was, early on in their relationship, he could not fulfill requirements to gain acceptance. His quest for Daisy is often touted as the embodiment of the idyllic American Dream. This dream is often something that can't be grasped, but within America, the idea is that if you work hard enough for anything, you can attain it. This work Gatsby does, although he encounters struggles with morality and time.
Gatsby deems Tom the enemy. Once he realizes Daisy is married, he invents the idea that she never loved him. This figment of his imagination plays out as Gatsby's affair with Daisy becomes a reality to Tom. Gatsby, for a short time in chapters 6 & 7, earns Daisy's favor as a result of her being in a poor marriage in the first place. This creates tension between Tom and Gatsby. This results in the questions of who is going to be in which car on a trip to New York. The fact that Daisy got into a car with Gatsby determined a victor for a time, but the brutal murder of Myrtle through the vehicle of the car signifies also the death of this conflict. Daisy from this point on chooses Tom although Gatsby doesn't really know it.
Gatsby's life conflicts all seem to stem from or involve Daisy somehow. His conflict with her seems to be that no matter what he tries, he cannot fully attain her. For a time, he is convinced that he has her, yet, she did not divorce Tom.