Can morality exist without religion?

Do most people really understand justice or Morality?
Evil people don't understand justice nor do they progress beyond feudal morality, but those who follow the Lord understand completely.
(Proverbs 28:5 NLT)

I think for an awareness of justice you have to first look at, and then past the court system.
Justice means righteousness, lawfulness, and moral rightness, the quality of being true or correct, the moral principle determining just conduct. So justice has a moral quality. It contains a concept of what is right and, therefore, must also have a concept of what is wrong.
Only people who understand these concepts of right and wrong can understand and administer justice. By extension, only those who follow the Lord can understand justice and morality. Why? Because they subscribe to the foundational laws of the one who created them.

Thus, as this proverb points out, evil people don't understand justice nor morality. Because they refuse to subscribe to justice's moral underpinnings given by God in his Word, they are left to discover their own truths. As a result, many conclude that there is no truth. Others conclude that everyone can have different truths. Both perspectives are hopelessly doomed. And as a building without a foundation will crumble, so justice can never be served without the foundation of right and wrong as given by God in his Word.

A godless justice can be bent to serve any means, financially or morality convenient at the time, meaning that morality would basically cease to exist, and the world becomes feudal, with modern technology, a feasible possibility in the forseeable future.
With God's word, comes absolute authority in god's law.

If a child was cought stealing you could tell him not to do it again because it is bad to do that.
Now, if you tell that same child not to do it again, because Jesus said so in his laws, that god laid down for all of us to follow, then he can rely (much as children do on parents) on the fact that there is a higher authority that will correct and guide him, and he will understand they why (moral) as well rather than just the legal aspect of things.
.