Consider the following debate on BBC's The Big Questions: https://youtu.be/UisxYorDNr4.
Seen it? Done? Good job. Now let's address the question.
So here's the thing. If by moral is meant knowing good from wrong, then the answer is: No, we don't. Man operates under duality consciousness and is therefore inherently aware of good and wrong (as it is of pain and pleasure, sadness and happiness, anger and delight, and so on) as it is.
However, we do have a propensity toward doing good as we are spiritually evolving (as opposed to devolving) beings, i.e., for example, we're not the same persons as we were during our teens. So all else being equal, the tendency is for us to do the right thing. (The wrong turns we take in life or the bad things we experience in our lives serve a purpose as well: they're the source of life lessons, the stuff that will catalyze the entire process.)
The process will eventually result in non-duality consciousness, i.e. a state of mind wherein the notion of good and wrong ceases to exist. And so here's the deal: whether or not you're religious, you're made subject to this process like everyone else and like everyone else you're being carried ever forward. Meaning, everybody will 'get there' eventually. Seeing things from this angle, I like to think that our one job in life is just to... live our lives, and that's all there is to it. To just live.
