Yes, if you can solve the problem of individual accountability to the morality system. But that's a very big "if".
A morality structure, religious or otherwise, consists of two elements: rules and accountabity (to the rules).
There are already examples of non-religious attempts of a morality system. Most political structures are based on this concept. Allowing right or wrong of an act (the rules) to be determined by the populous, or by it's ulitimate contribution to the culture/community, or whatever.
But, like the moral precepts provided by religion, morality of an "objective" system can only take hold when there is individual accountablity. Unfortunately, there is a significant percentage of any human population that will selfishly ignore "moral" precepts for personal goals.
Religion handled this through threats from "above", either through direct intervention (wrath of God, Karma) or denial of reward. Political structures do so with a brick and mortar approach, literally, threatening jail or fines. Smaller communities do so through banishment or shunning.
The flaws of these traditional accountability systems are obvious. Someone who doesn't believe in God or Karma isn't threatened by that punishment, and political systems can't make you accountable unless they catch you.
This gets worse. For those who do assign responsibility for their accountability to an "outside force" such as God or the legal system, getting away with an act that is moral questionable actually enforces their feelings of entitlement to that act.
Create a succesful structure of individual accountability, combine it with a structure of morals or laws that are fair, community-based, and able to improve, and you'll be well on your way.