One of the themes of Fitzgerald's novel relates to the idea of moral corruption. There is a specific idea that personal moral responsibility is sometimes optional, especially for those in a position of power and wealth.
There are no spiritual values in a place where money reigns... (eNotes)
Among the characters, Tom is a good example of a person that feels he is entitled to act however he wants without regard to either morality or his affect on the lives of others.
They were careless people, Tom and Daisy--they smashed up things and creatures and then retreated back into their money or their vast carelessness or whatever it was that kept them together, and let other people clean up the mess they had made. . . . (The Great Gatsby)
Tom cheats on his wife, insults the husband he is cuckolding and, finally, is instrumental in casting blame for Myrtle's death on an innocent man (aware that this will put the innocent man, Gatsby, in mortal danger). Tom has no moral fiber.
His personal history is important in the consideration his lack of morality. As a man born into money, Tom has no connection to work and has no financial accountability. This lack of accountability parallels his lack of moral accountability. His money insulates him from such considerations.
Regarding contemporary culture, we might argue that the noise of society serves to occasionally insulate individuals from moral obligation and accountability. Living in large populations and spending time in isolated social units, we may tend to feel divorced from others in our society and not involved in any moral/communal contract with them.
However, the ways in which you draw connections between Fitzgerald's novel of moral corruption and the society you live in will depend entirely on your own conception of your society and its culture.
Arguing that money remains central to today's system of values should be rather easy to do convincingly.
You might argue that people today are willing to leave problems they have created for others to clean up, as you suggest, because they simply lack maturity. There is an argument to be made that the culture of the West (in its television and film content, its pulp fiction, its magazines, its advertisements, etc.) does not expressly encourage or reward maturity. Your views and opinions, of course, will have to determine your agreement with such an idea.