Whew! That's a pretty complicated question. Let's break it into a few manageable pieces:

African Slavery. Strictly speaking, Davis's claim is accurate because Europeans did little actual slaving in Africa. David Eltis's Rise of African Slavery (2010) is an excellent study of why we shouldn't treat early modern slavery as a strictly Eurocentric affair or as something that white people simply did to black people. While the unfortunate souls stolen from the African interior suffered immeasurably from the early modern slave trade, some Africans on the coast and all along the trade routes also benefited from it. And while the European participants in the slave trade (including their counterparts in the Americas) should by no means be exculpated from their unconscionable complicity in dehumanizing their fellow human beings, they overwhelmingly participated as traders and consumers of slave labor, rather than as the slavers who "produced" slaves through warfare and kidnapping. So the claim, as stated, is technically accurate.

Barbary Slavery? But regardless of how you assess European involvement in the African slave trade, before it to the Barbary slave trade, it's important to parse just what Robert Davis might mean by the Barbary slave trade. Wikipedia defines this as "white" slaves being sold into Africa between the 16th and 19th centuries. This article gives short shrift to the trans-Saharan slave trade, which Wikipedia also confusingly indexes as the Arab slave trade. I don't think you can get a good picture of what was going on in the grand scheme of things unless you consider all these systems together. (Also, I'd remind you that the way we tend to organize the world into races and continents isn't necessarily the way that premodern people would see the world or their fellow human beings.)

Trans-Saharan Slavery. The trans-Saharan slave trade was in continuous operation from antiquity until the 20th century. A high estimate for the years 1400-1900 is 14-15 million Africans. A more conservative estimate, but spanning the longer years 600-1500, is 5.5 million slaves. I'd recommend the superb overview by Anne Haour.

European Slavery. As Michael McCormick has argued, European slave markets operated in the Mediterranean (and so would be implicated in the Barbary slave trade) continuously between antiquity and the early modern period. In fact, he asserts that Europeans selling other Europeans into the slave markets of the Mediterranean was the impetus that established the basic economic structures that would come to be known as "Europe" (or at least "Western Europe"). McCormick thinks that Venice especially thrived off of the slave trade, and once bishops prohibited selling your fellow Christians as slaves, the Venetians fought hard to protect their rights to the slave trade coming in from their pagan Slavic neighbors. It seems like selling you neighbor into slavery was something akin to "priming the pump" of premodern economics. So just early modern Africans sold their own to European traders, it looks like Europeans were enslaving Europeans, and then exporting them to the Barbary shore.

Arab Slavery? So we've got about a millennium or so worth of slaves coming into the Mediterranean from both "black Africa" and "white Europe." Where were they going? That's the million-dollar question. The assumption has been that they were going into Islamic lands, but at least for the early medieval period, this assumption is based on some dark assumptions that Islam is a bad religion and somehow needs slaves. Even if you can swallow these appalling claims, they depend upon a contingent claim that everybody living in Islamic lands subscribed to this particular brand of Islam. The sources suggest otherwise.

Bringing It All Together. So we've got a lot of evidence for slaves coming into the Mediterranean from both north and south continuously from antiquity into the early modern period. Most seem to have been destined for North Africa or the Middle East, but we don't really know why. Numbers are also an issue. I haven't seen any compelling estimates for how many slaves were sold south from Europe, and McCormick's arguments become very impressionistic after the early middle ages.

Comparisons We Can Make. We can do a bit better with the trans-Saharan trade. If we take the cautious estimate of 5.5 million trans-Saharan slaves for the entire middle ages, this pales in comparison to the 7 million or so Africans exported to the Americas in the 18th century alone. If we take the more ambitious estimate of 15 million trans-Saharan slaves during the early modern period, this would put it roughly on par with the trans-Atlantic slave trade.

Conclusions. What this suggests to me is that the scale of early modern slavery vastly exceeded the scale of all previous African and European slavery. Davis's claim is evocative, and it makes the good point that we need to consider trans-Atlantic slavery in broader contexts of slaving practices. But it would take some fancy footwork to support the claim that slavery caused as much damage on the coasts of Europe, even looking all the way back to the time of the Roman Empire, than it did to the African interior during the few centuries of the trans-Atlantic boom.

Here's a link to the research page from Ohio State announcing his book (it's a very old web page).

http://researchnews.osu.edu/archive/whtslav.htm

I'm not finding that claim there, so I'd be curious to know where you sourced it from (I'm not doubting you, just wondering if this is something where his research has been exaggerated by someone with an ax to grind).

EDIT: It's totally there, just phrased differently. I didn't read good.

In any case, the significant quote from that site seems to be this:

Over the course of four centuries, the Atlantic slave trade was much larger – about 10 to 12 million black Africans were brought to the Americas. But from 1500 to 1650, when trans-Atlantic slaving was still in its infancy, more white Christian slaves were probably taken to Barbary than black African slaves to the Americas, according to Davis.

Now, there's some interesting framing going on here -- notice the non-parallel "black African" and "white Christian" -- and also the fact that his book is called "Christian Slaves, Muslim Masters: White Slavery in the Mediterranean, the Barbary Coast, and Italy, 1500-1800."

His claim seems to be that between 1530 and 1780 there were about 1 million to 1.25 million people enslaved by the Barbary states (his methodology is essentially done by calculating a "replacement rate" for how many people would have to be enslaved per year to fill what he has researched to be the "normal" number of slaves in Barbary).

Now the issue becomes how we would define "exceeded the damage done to the African interior." First off, are the European coasts and the African interior comparable places? What do we mean by "damage?" Why exactly would he (or whoever made that statement) pick those two spots, when most of the African slave trade was coastal (although supported by raids into the interior)? You can see that the old "which was worse" claim gets very hairy very quickly.

Notably, this quote is not actually a quote from Davis himself, but comes from an OSU press release for his book, Christian Slaves, Muslim Masters: White Slavery in the Mediterranean, the Barbary Coast, and Italy, 1500-1800. So this is not a claim that Davis himself is making, although this quote has been picked up and attributed to him directly on Wikipedia, from whence it has become "fact."

His book, in fact, expressly does not make this claim, quite the opposite:

Certainly [the Barbary raids'] cumulative devastation fell far short of that produced by the slaving wars that were even then beginning to ravage the interior of West Africa; nor did they provoke anything like the net population decline that would later afflict the African states. (p.24)

This is because, as Davis elaborates on, the sort of sophisticated, market-driven slave raiding/trading networks which reached into the interior of Africa never developed in the Mediterranean slave trade. Davis explicitly contrasts the "pre-modern" raiding style of the Barbary slaving with the system that fueled the Atlantic trade. He attributes this to both the difference in goals of those two slaving systems, but also with the fact that powerful European states like Spain, France, and some Italian polities, were able to mount effective defenses against slave raids. While this did lead to poorly quantified secondary costs in terms of "lost investment capital, frustrated development projects, or abandoned settlement," we are really no longer dealing with the original question at this point.

When Davis does deal directly with "depopulation" of European coasts, he focuses almost entirely on Italy, which he notes as being the hardest hit European region, in part because of proximity and history, but also on account of its disunification at the time. He relays the alarming fact that certain coastal towns in Southern Italy suffered population declines of 40-80%, but notes that this was a secondary effect to slave raids, as the coastal population migrated inland to larger towns and cities. Or as he puts it:

... one can well imagine how little appeal there must have been in continuing to eke out a life that had always been impoverished but was becoming increasingly dangerous as well -- especially with the booming baroque capitals of Rome, Naples, or Palermo acting as powerful magnets for poor emigrants from the hinterland.

So again, that quote is not actually something Davis said and does not accurately represent the positions he puts forward in his book. From my own perspective, I would attribute the popularity of the quote as a demonstration of second option bias (good explanation from our own /u/NMW). It fits the contrarian narrative of being able to say, "Yes, Black people had it bad, but do you know that White people had it just as bad?" which relies on anachronistic groupings and ignores the particulars of those events (and the facts themselves).

The widespread material and research on the Atlantic Slave Trade is not some conspiracy to whitewash (heh, "white"wash) the slaving done by the Ottomans and Barbary states. It is a because widespread enslavement in Africa did happen on a much larger and organized scale resulting in wide-ranging and long-lasting repercussions ranging from colonial and post-colonial exploitation to the development of pseudo-scientific approaches to human diversity and entrenchment of racist ideology.

e: spelling