The idea that greed and the personal ownership of private property are behind much of the injustice in the world needs to be examined carefully. For example, greed is not necessarily the same as the ownership of private property. One can own property and not be greedy, just as one can be greedy and not own private property. Similarly, one can be greedy and own private property, and one cannot own private property and not be greedy. Greed and the ownership of private property, in other words, have no necessary connection to one another.
By the same token, greed can sometimes definitely cause social injustice, but the two need not be connected. Some greedy people, ambitious to make as much money simply for themselves as possible, have started businesses that have also benefitted investors, workers, and society at large. Donald Trump might be a case in point. He is perhaps not the most appealing human being, but his ambition has indirectly helped others.
The idea that private property should be eliminated in order to eliminate social injustice is an idea with a very bad track record. Attempts to establish communism in various countries in the twentieth century usually ended in disaster, not only economic disaster but often moral disasters involving huge losses of life. Thomas More, in fact, is probably not imagining communism in the modern sense at all in Utopia but is instead probably drawing on ideals of monastic communities, in which, theoretically at least, private property did not exist.
Other historical instances in which private property has been totally eliminated (at least for some) have often involved some kind of slavery or imprisonment. Slaves could be property, but often they owned almost nothing themselves. Prisoners are usually allowed to own very little private property.
A strong case can be made that ownership of private property can lead to greater social justice than past attempts to eliminate private property have produced. Ownership of private property is one guarantor of freedom. It is also often the precondition for economic innovation and economic progress. Competition between owners of private property can often foster economic growth and thus benefit society at large, especially if ownership of private property is widespread. If private property is held in too few hands (as it tends to be in societies rooted in slavery), then only a few private property owners benefit.
The best system, then, would seem to be one in which ownership of private property is common at every level of society and in which owners of property have incentives to compete with one another.
In the words of President James Madison,
As a man is said to have a right to his property, he may equally be said to have a property in his rights.
In other words, it is very difficult to separate private property and human rights.