Where I disagree is this idea - whether you intended it or not - that non-absolute morality is somehow arbitrary, or worth less than absolute morality.

I don't agree with that: I believe that torturing someone is immoral because humans dislike being tortured (if you're in that 1% who doesn't mind torture, it's not immoral - torture away). You reduce that to "most humans don't feel positive about being tortured". But that's not a choice that 99% of humans made: it's a fact of human biology. Humans don't choose to feel pain when someone rips out their fingernails. You can say "disliking pain is arbitrary". It may be arbitrary in some sense, but not in the context of humanity, and we're all humans.

If you want to say that this kind of "moral thinking" isn't "true morality" because it's not absolute, fine. That's valid stance to take. But whether you call it morality or something else doesn't make it less valid or less useful. And to suggest that humans should not take our innate likes (warmth, safety, respect) and dislikes (pain, insecurity, lack of respect) into account when we interact with each other simply because human biology isn't "absolute" is - pardon me - fucking insane.

Natural Law is NOT a new idea, really. Nor is it foreign to religion! St. Thomas of Aquinas is the father of Natural Law -- almost a millenium ago now -- and a big deal in Christianity. Natural Law is a big deal in the Catholic Church, for example. One might say that St. Augustine vs. St. Aquinas is just a religious expression of the Plato vs. Aristotle.

You'll find plenty of well-read, well-educated, very religious people who take a Natural Law approach to morality and who would agree that what happens in private between consenting adults is almost entirely moral (with some limits, such as perhaps that giving one's life up consensually is not moral). (I myself am agnostic, but still, I take a Natural Law approach to morality.)

Of course, there are schools of thought that reject Natural Law, but you must at least acknowledge Natural Law as a concept. And you should at least accept that Natural Law works for others even if not for you. You have to co-exist with the rest of us.

(Us noticing that other animals have empathy is fairly new though, I think.)

You say something is wrong, I can say you saying that something is wrong is, in itself wrong. Ultimately, it either doesn't matter at all (we're all just talking), or enough other people agree with us that there is some social consequence.