Asking if morals will exist outside of religion is the wrong question. Of course, morals will exist. The real question is, whose morals will we follow?
Morals are the rules a society uses to conduct itself in order to function in a stable and healthy way. If a man or woman rises into such a place of power that they can impose their morals and values on others, then we better hope they are ethically sound people who care about our wellbeing. Hitler had morals that seemed right to him. Those did not work out so well for others, and people suffered greatly.
A defined morality that originates outside of man solves the problem of morality being established by whoever wields the biggest stick. I am not offering this as proof of God, btw.
I do, however, agree that this sense of hoped-for "cosmic fairness" contributes to the origins of many religious concepts. Whether God exists is for each to decide.