I'm not sure exactly what you mean by innate or inherent. If you mean that human values could be present from birth, then I'd say probably not. Sure, there's an evolutionary argument to be made about self-preservation and even altruism, but I think this won't get anybody very far toward something I'd recognize as "human values."
I would suggest that the variation in values suggests that they are to a large extent cultural – and thus acquired, though I don't know if 'taught' is exactly right.
Your question about innate or acquired values opens onto the question of whether values can be universal and whether there is any ultimate ground for them; and a response like mine can be misinterpreted in a couple of ways:
First, people may think that acknowledging cultural variation in values leads to some kind of relativism, and that relativism is really nihilism. (Think of the effect that Herodotus' examination of other societies' morals is said to have had in classical Greece.) This deserves a longer response, but I think that it's a mistake to de-value values because we recognize that they aren't universal.
Second, people might imagine that recognizing values as cultural means that arguments about values cannot be held, or that the best we can do is try to extract some universal core of morality by comparing different cultures. Philosophy and religion offer ways of grounding values, like Kant's categorical imperative or divine commandment, and if these no longer appear to be tenable as absolute grounds, then it's tempting to think that these attempts are misguided, or even that we can't argue about morals. But, again, I would say that this is a mistake, that ethics and values depend on just such a conversation, even if it's a conversation that may never stop at a firm resting-place. In fact, if I were to try to produce an argument to support my suggestion that values are cultural, it would probably involve looking at the way that this cultural conversation (carried out in literature, philosophy, religion, and generally those things studied by the disciplines we call humanities) produces values and sources of value, and especially ways of sustaining the human as a value-term in the face of historical changes.
