One answer is what is often called "enlightened self-interest". The Golden Rule is basically a statement about enlighted self-interest.
If I can convince someone that abortion is the killing of a human, then I can go on to say that if we allow the killing of some humans, then what is to say that we won't someday allow the killing of humans like him.
This is one of the basic arguments against euthanasia. You can't get a grown person to worry about being the victim of abortion. But you can get that person to worry about being the victim of forced euthanasia: at some point he may worry that it is "too" expensive for his family and/or society in general to keep him alive as long as he wishes to. Here is a person, religious or not, that you could convince to simultaneously oppose both abortion and euthanasia just out of self-interest.
He may be willing to fight against both while he is young as insurance against the practices becoming normalized when he is older and unable to fight.
Democracy and free markets are all about enlightened self-interest. The best form of government would probably be a monarchy led by a great and wise leader. Unfortunately there is no such thing as a leader that is great enough and wise enough to lead a country as large as the US. So instead we settle for the half-baked compromises we get under Democracy.
Similarly the most efficient economy would be one where the best brains to tackle a particular problem were assembled with the proper capital to solve the problem as quickly and cheaply as possible. Instead we let anyone who wants to try and solve any problem they can. This leads to massive amounts of money being spent by people who never solve the problem. Still this ends up being cheaper because we haven't found a way to identify the people who are the best at solving particular problems.
Similarly with moral issues we let people come to different conclusions and we see what works and what doesn't. All the statistics suggest that those communities that have strict gun laws tend to have more crime. If we could only get the media and the public in general to respond to this data then we could get gun laws relaxed throughout the entire country. And this without ever needing to invoke a higher power.
There are more and more people that are not only blase about God, but are downright hateful. And this hate doesn't come from something they feel God did to them, but because of all the bad examples they have from all the scum that have done evil in His name.
Mortimer Adler was a neo-Aristotlean virtue ethicist all of his life. He did great work bringing the commonsense of Aristotle to the masses and the elites. In the end he became a Christian, but he did a lot of good even before finally coming into the Church.