Or vice versa. From a cross-cultural perspective, once one sheds the usual prejudices, it doesn't appear that non-Christian societies are, on the whole, any worse places in moral terms than Christian ones.
Keep in mind that morality is a subjective thing, and that the Greeks, Romans, Germans, Celts and Slavs all had their own moral beliefs about what was right, wrong, appropriate or taboo.
You're right that allmost all lasting society develop at least some kind of system of morals and standards which keep the society stable. And yes, there are usually good things about those morals.
But that still means that moral standards can differ between societies for the worse - some practices, like honor-killing, female circumcision and forced marriages, are perfectly allright by the moral standards of some societies.
And the introduction of early Christian moral standards in the Roman Empire indeed have changed a few things for the better, such as the abolishment of gladiator-fights, crucifiction and the branding of slaves, like Merryprankster had already pointed out.
On the issue of slavery, I believe it would have died out, Christianity or no Christianity, once it was no longer economically sensible, probably to be replaced by a form of peonage not unlike that of serfdom in OTL's medieval Europe.
I agree, I'll have to check some details on this, but I'm pretty sure that after the Roman era in the West, slavery was never abolished, but just fell out of use. Only with the conquest of the New World was the practice of slavery revived, first with the Native Americans and some other groups (Polynesians, etc.) and then finally with black Africans, because they were resistant to the European diseases, as well as for being sturdier and better adapted to the tropical conditions.
Whenever Europeans turned to slavery, that was mainly for purely practical and financial advantages of slaves over free laborers in those circumstances. But as soon as slavery became impractical, it quickly fell out of use and disappeared.
That's why we hear so little about the Native American slaves that the Spanish had.
The fact that slavery was reinvented in the early modern era by thoroughly Christanized societies is evidence enough that Christianity can and does live comfortably with the mass enslavement of human beings whenever convenient.
Keep in mind that Christianity is very sensitive to syncreticism and absorbing local customs and ideas, and that religions and religious communities
can be affected by local ideas and customs, even to the point that common customs contradict the actual teachings of the religion.
For example:we all know that Islam quite strictly forbids the recreational consumption of alcohol, yet this doesn't prevent many millions Turks from appreciating a good glass of whisky or raki. And there are much worse examples than this, mind you!
So if you say that Christian societies can live comfortably with such rather atrocious practices are going on like slavery as seen in the Americas or the ruthless exploitation of laborers during the Industrial Revolution, then - yes, you're right about that.
But Christian societies don't neccesarily stick to what the actual Christian religion and scriptures teach, so Christian societies can actually live quite comfortable with things that absolutely don't go with actual Christianity.
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One footnote: it really suprises me that nobody here has mentioned anything about Neoplatonism yet.
Neoplatonism was the philosophical aspect of the late Greco-Roman pagan religion, and unlike the usual everyday traditional religion, it was spiritually very strong. In OTL, traces of Neoplatonism survived until the very fall of the Byzantine Empire to the Turks, and a number of muslim sects in the Middle East have been seriously influenced by Neoplatonic thought and concepts.
In fact, the last pagan Roman emperor, Julianus Apostata (Julian the Apostate), was a passionate adherant of the Neoplatonic philosophy.
It is my guess, that without Christianity, the old traditional pagan religion would be slowly transformed and deepened by Neoplatonism, and especially if/when the old traditions start to lose their religious significance, Europe will propably be dominated by a developed form of Neoplatonism in which the old traditions have relatively little importance.
And one more detail, a strong presence of Neoplatonism (which would be quite likely in a no-Christianity TL, hence the other major rivalling sects of Neoplatonism were usually pseudo-Christian Gnostic sects, who would also be absent from a TL without Jesus) would also quite likely contain the spreading of Zoroastrian ideas and beliefs in the Roman Empire.