Achieving adulthood or manhood in America is all about working hard to earn a good living and provide for oneself and one's family, no matter what your circumstances were when you started. It is all part of the American dream. In fact, if you ask young men today what it means to be an adult or a man, a lot of them will say that graduating college, getting a job, starting a family...all of these things are signs of maturity. In Dave's time, it was similar. Being a man meant working hard to earn your way.
In other cultures, there are other types of ceremonies and rituals that a young man has to undergo in order to achieve manhood; for example, things like spirit walks. However, for Davey, to prove his manhood he only need to take responsibility for his actions, and work hard to pay for the mule, Jenny, that he killed. Dave makes many mistakes, but he is right about one thing: getting respect from others will help him to become a man. However, he thinks that having a gun will get him that respect; he misses the important lesson that working hard, taking responsibility for his actions, and doing what is right is what truly earns respect.
In America, working hard and achieving success has always been a part of being an adult, and earning that status of maturity. It ties in closely with the American Dream. I hope those thoughts helped; good luck!