After the fall of the Roman Empire, we have a gap of several centuries during which little history was recorded, though we do have a few anti-slavery documents from that period. For example, the writings of Saint Patrick are certainly against slavery. When the historical record resumes around the 600's and 700's, we find many Christian monarchs and leaders, such as Saint Bathilda, working to abolish slavery in Christian territories. Within a few centuries the practice of slavery through most of the parts of Europe where Catholics were dominant was gone, although it was still legal to use prisoners of war for forced labor. Hence western civilization became the first civilization to abolish slavery for moral reasons.
It is, of course, true that after European nations began colonizing the Americas they introduced slavery there and brought millions of slaves from Africa, and that will forever be a stain on history. However, for the purposes of this argument it should be noted that the Catholic Church was always opposed to slavery, as indicated, for instance, by the Papal Bull
Sublimus Dei issued in 1537. The Church did not have the power to make laws in the Americas, however. It's also worth noting that slavery persisted in the Arab world and much of Africa long after it was abolished throughout the Americas, and in many places stopped only after European colonialists took over.
Perhaps if we're going to bridge this divide, we first need to agree on what the phrase "human rights" means. To me, it means at a minimum the trio of life, liberty, and property, and an announcement that the government can neither remove these three things from any person extra-legally and also that the government has a positive duty to protect these three things from any threat. Would you agree with that definition?
Under that definition, plainly the code of Hammurabi contains no trace of human rights, since it does not put any limits on what the government can do, but only on what ordinary people can do. I'm not aware of any ancient Middle-Eastern nation where there were limits on what the king/despot/emperor could do. They were absolute rulers, nothing less. As for the ancient Greeks, I'd have to know who you were talking about before I could discuss whether there's any notion of human rights in their writings. The most famous political writing of ancient Greece is Plato's
Republic which conceptualizes an ideal nation where the government has absolute power to exterminate those its views as inferior, censor anyone for any reason, and so forth.