Perhaps if the wealthiest among us suffered more equally the pain of fighting wars and of medical care insecurity, they would invest more heavily in world peace and healthcare, and less on Angry Birds and luxury taxi services. This is why equality of quality of life matters -- to keep us focused on each others' wellbeing.
We do have virtually equal quality of life in the US. A rich person can have ten cars, but he can't drive ten at the same time. It is a left-wing cliche that the burden of fighting wars falls upon the poor. In the real world we live in, the military is all volunteers and mostly middle-class, not from the poorest demographics. And pretty much everyone in the USA has access to state of the art care. You cannot buy your way to the top of an organ donor list or something. Certainly there is a difference between rich and poor, but you have not shown that it is an unjust difference or that it is harmful to the poor person.
