Yes. Medieval western Europeans absolutely continued to enslave people after the disintegration of the Roman Empire, especially in the lands around the Mediterranean. I'm not sure why you're specifying "white slavery." Not everyone in medieval Europe was white, and not all enslaved people in Europe were from Europe. There are absolutely times in the history of pre-Atlantic enslavement when skin color mattered to the enslavers...but maybe not in the way you think. The medieval slave trade was a circum-Mediterranean and Black Sea horror.
The most famous/infamous example is surely the phrase that English chronicler-historian Bede puts into the mouth of Pope Gregory I (r. 590-604). According to Bede, Gregory noticed some enslaved boys being sold in a Roman marketplace whose skin was much paler than he had seen before. When someone told him they were "Angles," he responded, "Non angli sed angeli"--Not Angles but angels. Notice how unconcerned Gregory--and more to the point, English monk Bede--is with the concept of enslavement.
600 is awfully early, though, so let's jump to some later examples.
Despite 100% of reddit's empathy for people in warfare lying with the soldiers who committed all the atrocities, it was absolutely horrific to be a civilian in the way of a medieval army or raiding party. Living in a city that was sacked or being caught up while trying to flee, or simply having a village in the wrong place at the wrong time, most likely meant men would be killed; women and children (boys and girls) would either be murdered or sold into slavery/sexual slavery, and probably the soldiers would rape them on the way.
John Gillingham has argued that, well into the high Middle Ages (1000-1200), enslavement of defeated women and children wasn't just a side effect of medieval warfare--it was a major goal.
This practice gradually died off in western European warfare as more and more of the continent was converted to Christianity. Eventually, canon (Church) law forbid Christians to enslave other Christians...but as you should expect, that is not how things turned out. William Phillips, for example, identified a handful of Eastern Orthodox enslaved people in medieval Spain.
The examples I really want to talk about, though, come from the late Middle Ages. By this point, the presence of "exotic" enslaved people at the courts of the high nobility and royalty were an impressive status symbol. It's not wrong to think of the enslavers as treating their fellow human beings in the same way they treated animals in their menageries, just dolling them up in elaborate clothing. But elaborate clothing and sitting prettily at court was by no means the fate of most of Europe's enslaved people in the late Middle Ages.
I'm going to borrow from an article I wrote for a medieval history magazine awhile back, which I've previously cited on AH as well:
For enslaved women forced to serve as wet nurses, the situation was even more demeaning than that of the municipal nurses. Since 1179, canon law had forbidden Christians from hiring Jews and Muslims to nurse their children (although the frequency of reiterations—from Christian and Jewish authorities alike—points to the frequently-informal shared breastfeeding duties among Christian and Jewish women neighbors). And yet from the thirteenth century into the early modern era, slaves in the Christian Mediterranean were almost always purchased or captured Muslims. To Christian parents, the solution was simple: forced baptism.
And even beyond this indignity, evidence concerning baptized Muslim wet nurses indicates the growing tendency to classify and judge people by skin color. The humour and heat imbalances in darker-skinned women made them bad mothers, according to medical authorities. And since wet nurses were understood as little more than biological appendages of real mothers, over and over, slave purchase and rental contracts demonstrate a keen preference for light-skinned baptizatae of nursing age.
Yes, rental contracts. Enslaved women pressed into service as wet nurses weren’t always just serving the family they at least knew and were comfortable with. Owners might hire out a baptizata as a nurse for some extra cash, or sell her altogether if the price was attractive enough.
And if no wet nurse was available? The brutal, systematic, and endemic sexual exploitation of enslaved women in Iberia could be—was—turned to economic use. When the foundling hospital of Perpignan was so financially overwhelmed in 1456 that it appealed to the city for extra money, the rectors made it bleedingly clear that aristocratic men bringing in their own bastard children were to blame. One man in 1400 Barcelona openly admitted sending away his own child so the baby's mother, an enslaved woman, would be free to nurse his legitimate heir.
And therein lay the cruelest blow of medieval wet nursing practice. Medical thought and popular religious teaching alike forbade women from nursing more than one child at the same time. All these wet nurses, all these “mothers who weren’t,” were mothers. They were mothers who weaned their children too early and quickly in order to make some money for the family; they were mothers who lost a child in infancy and had milk but no one suckle; they were mothers sold away from their newborns forever.
It absolutely did, at least for a while, and at least in some parts. While the Catholic Church had views on slavery, its opprobrium only really extended as far as owning fellow Christians. Apocrypha has it that Pope Gregory was inspired to order the conversion of England after witnessing the plight of Pagan English slaves in a Roman slave market, which at the very least posits the existance of a pan-European slave trade in the mid-late sixth century.
The conversion of England may have begun in 597, but slavery remains prominent in England well into the 11th Century, and possibly into the early 12th. The slave trade is a major economic impetus for the English and slaves form a significant proportion of the population. By the time of Domesday Book in the 1080s, approximately some 10% of households listed are slaves, and most settlements have at least one slave household. Usually when reading Domesday, a "household" is assumed to be around four people, although there is debate as to whether a slave "household" represents just a single slave, or indeed a whole family. Therefore we can estimate that slaves comprised anywhere from 2.5% to 10% of the population, and this two decades after the Norman Conquest, since when slavery had fallen strongly out of fashion and been in steep decline.
The precise nature of Anglo-Saxon slavery is unclear, but it seems to be the case that slaves basically formed a social caste used predominantly for agricultural labour. Slavery was covered fairly extensively in contemporary law, and slaves even had some protections. The majority of slaves would have been Welsh, and by the tenth century, these slaves were also being exported to the Hiberno-Norse in Ireland from the ports of Bristol and Chester. In sixth-century Kentish law, thieves caught in the act and killers who could not pay their weregild were liable to be sold into slavery, but this had to be overseas. By the seventh century in Wessex, however, the laws of Ine of Wessex strictly forbade the selling of any Englishman out of the country. By the reign of Alfred in the late ninth century, the law was more explicit on the Welsh nature of slaves. Slaves were protected under the law as any other person, although their weregild was lower than freemen and liable to their master rather than their relatives.
Slaves were often emancipated in wills, but could also be transferred as property. The tenth century will of a Wynflæd, for example, emancipates one of her slaves and leaves her a small package of land, but also leaves other slaves including 'a spinning woman and her wheel' to her daughter. Under the laws of Alfred's Doomboc, any slave that was forced to work on a Sunday would be freed, even if they themselves weren't Christian. In a similar vein, any child born of a liaison between a master and his slave was freeborn, and had to be provided for, unless the father disowned the child, in which case he forfeited all rights to the child or their weregild, and had to provide a relatively substantial stipend to the mother.
More of course can be said, but this older chain from /u/gunlord500 and /u/textandtrowel might be of interest for you!