This question essentially boils down to two main factors: logistics and religion. Slavery actually persists in parts of Medieval Europe at least until the 11th Century, but on nowhere near the scale as it was in Rome and partially, yes, this is due to Christianity. While Christianity doesn't forbid the owning of slaves, it does forbid the owning of other Christians as slaves. As the boundaries of Christendom expand in the early medieval period, so the available pool of slaving targets also diminishes. Supposedly, Pope Gregory I ordered the Augustinian mission to England in 597 as a direct result of witnessing the plight of Anglo-Saxon slave boys in a Roman market, and wanted to spare any further English the same fate. For the Christian English, the Welsh and Irish were often fair game for slaving; for the Holy Roman Empire it was the Pagan Wends and Slavs along the Eastern frontier.

Another significant factor is logistics. I've talked elsewhere about how the decline of the Roman Empire and the concurrent economic collapse brought about a decline in building in stone, a return to subsistence agrarianism and changes in weapons and armour, and slavery is no different. A slave economy like Rome's requires staggeringly vast amounts of manpower, which are kept fed by Rome's ceaseless border wars feeding a steady supply of captives back to the interior. For all of its historical problems, consider the film Gladiator as an illustration of the importance of the Roman logistics network: Maximus is on campaign in Germany, returns to Hispania, is sent as a slave to North Africa and ends up in Rome. In real life, the Roman network allowed for vast slave workforces to be easily disseminated around the Empire, and for supplies to easily reach them. As these networks decline, new polities and borders arise, infrastructure is less maintained and organisational structures collapse, it's simply no longer feasible to marshal such large workforces so easily, and also much harder to maintain them. Without a wider logistics network, it's also far less viable for large villas to form the basis of the agricultural system, and instead, variable crop small-scale local farming re-emerges in response for the need for local and reliable food supplies. Slavery does survive, particularly noticably in England, but at Domesday 'slaves' account for around 10% of the listed population. Anglo-Saxon society is heavily stratified, and based on a system of reciprocal patronage throughout. Slaves are just the bottom rung in a ladder that includes the 'semi-free', freedmen, ceorls and freemen of various classes.