Most if not all values people accept or reject in this life are relative, not absolute. And that is simply because absolute values do not exist in this life, in this world.
Values differ from one culture to the next, one point in time to the next, one person to the next. It is the differences that comprise the relativity of values.
Take murder for instance. Most cultures today would condemn murdering children as "bad" or "wrong". Nowadays most people share values relating to protection and nurture of children. But in many ancient cultures, including the Carthaginian and Aztec cultures, child sacrifice was regularly practised on a large scale (see image above). In the Aztec culture, thousands of children were ritually slaughtered to appease the god Tlaloc. In the Old Testament the god Jehovah asks Abraham to sacrifice his son Isaac, saying: “Take your son, your only son Isaac, whom you love, and go to the land of Moriah, and offer him there as a burnt offering on one of the mountains of which I shall tell you.” (Genesis 22:2-8).
But that was then, and this is now. Things have changed, for the "better" (whatever that means), haven't they? Today, all right-thinking people agree on what is right and wrong, don't they?
Well, actually, no. Here’s a good example: Mutilating the genitals of children, male and female, is still widely practised today. Some people think that genital mutilation is abhorrent, bad, wrong, evil. Other people think it OK, fine, even good. And yet another group of people believe that genital mutilation would be abhorrent, would be bad, would be evil if it were not a cultural practice, (and therefore not to be criticised or evaluated in any manner, shape or form, on any grounds at all, ever). Clearly, there is no agreement on the rightness or wrongness of genital mutilation. It is a relative not an absolute value. Just like the culturally determined value, with no rhyme or reason to it, under which some men wear neckties to work.
Most if not all people assess the worthiness or otherwise of their actions and thoughts by reference to rules or standards imposed by an authority or authorities external to the individual (eg God, gods, the law, the priesthood, the army, mom and dad, the headmaster, social worker, etc). But often, the external authority has feet of clay and acts or speaks in a way that is inconsistent with the values previously established by the selfsame authority. That's what we call hypocrisy. ("Don't do as I do, do as I say.")
The New Testament records that Jesus lost his temper with a fig tree and withered it when it would not produce fruit on demand and out of season. So, in the context of Christianity, unreasonable anger is not seen as bad, or wrong, or to be avoided. In other contexts, unreasonable anger is not valued so highly.
Some (very few) people have a strong enough belief in themselves consciously to reject the values of their culture and create their own values to live or die by. They can and often do refuse to submit to the values imposed by external authorities. Psychopaths are among those who are able and willing to examine and if need be reject the values of their culture in favour of their own 'home-grown', do-it-yourself values. Instead they use the compass of purpose to guide their behaviour. (Or so they telos!)
To illustrate, if my purpose is to travel from Paris, France to Berlin, Germany along the shortest possible route. For me to make that journey via Sydney, Australia would not be "wrong" or "bad", but rather, would not serve that particular purpose.
Another example: If I want to have a healthy respiratory system, then to smoke cigarettes would not serve my purpose. It would not be wrong or bad to smoke; it would be inadvisable given the stated purpose.
Another example: If I don't ever want to be punched in the face, then for me to punch someone in the face will not serve that particular purpose. Because by punching someone in the face I increase the likelihood of retaliation in kind. From a purpose-driven perspective, therefore, it would not be "wrong" or "bad" for me to punch someone in the face; instead, it would simply be an action that would not serve my purpose.
The answer, therefore, is no: we don't need God, or gods, or any external authority to dictate morality. The results produced by purpose-driven, values-free living are indistinguishable from those produced when values are imposed or dictated by an external authority. It's a bit like the "invisible hand" of market forces, but there's no time or space to get into that now.