That's just one of Paul's many confusions. (I criticize Romans in the two articles linked below.) What's innate in primates, due to evolution by natural selection, isn't morality but tribalism.
Babies do indeed have an inherent sense of empathy, which facilitates their bond with the in-group. But they also naturally develop a sense of fear of outsiders. Their moral instincts are rudimentary and have to be nourished with training, so much of morality is learned from culture.
As Lawrence Kohlberg showed, the universal kind of morality that religious people have in mind is learned, although it sprouts from those biological seeds.
So no, enlightened, universal morality isn't innate, but is largely taught to correct the tribalism that is indeed "written on our hearts."
You think God is needed for morality. Meanwhile, we see in animal behaviour and in the proto-moral instincts of babies how morality instead evolves from more primitive, tribal behaviours.
I take it you think nonreligious people leech their morality off of religion and God. But couldn't we just as easily say that religion leeches its moral standard off of biology and evolution? Or do you think God has his hand in all the mayhem of animal behaviour?
As far as I can tell, the moral standard derives from human reflection on the world around us, including our tribal nature.
