I believe this idea is refering to the natural moral law. For some Christians, it was God’s revealing of this “natural law” on Mt. Sinai (ten commandments) that caused it to become objective. The process of self-justification that stems from pride in the sinner would seem to inhibit the “objectivity” of any alleged natural moral law. Therefore - according to that logic - without God’s intervention, because we are sinners we could never discover the natural moral order on our own.
Another idea I have heard from apologists - which would seem to fall into this category - is the idea that without a God that actually exists, there are no consequences for bad behavior. As long as you can get away with things you know or do not know to be wrong, there is no reason to involve your conscience. If God does not exist, we all just turn back to nothing - or dissolve into the ocean of evolving particles of consciousness that concentrate around and within the higher beings. There is, according to this view, no reason to try to follow some sort of moral code if God does not exist. If you can get away with killing someone for selfish motives and it will have no lasting repercussions, there is no objective reason not to do it. Admittedly, most of us live our lives this way - we weigh the consequences of our actions and the possible outcomes. If we might be able to get away with something that is conventionally considered “wrong,” we will probably consider doing it - if only momentarily - if that would be to our advantage.
I think that moral objectivity is possible without explicitly bringing God into the picture. Ethical activity is based on respecting others. We respect other people, because we want them in our lives - because we love them and love requires attention and respect. Friendship drives us to develop virtues - virtues that are needed in order to live in harmony with the people we love. These virtues extend beyond the people we love however - for example, when I recognize that my friends must be treated justly, I also become capable of seeing that people I don’t know should also be treated justly, and finally that people I do not love or even hate ought to be treated justly as well. Obviously, when one has little or no personal virtue, when one has no true friends, the objectiveness of morality is a bit of a long shot.