I posted a related (though narrower) answer recently about how Roman freedmen viewed slavery. Broadly speaking, I believe free Romans would have passively accepted slavery. Acceptance of slavery was likely a rather mixed bag among the slaves themselves (who, at something like 1/3 of the population, should probably be considered everyday Romans too). The Romans were keenly aware that all their slaves wanted to be free, but the slaves who went free were almost by definition those who had accepted slavery and internalized their enslavers' values.

I haven't read Debt, but the passage it's quoting is from the Digest (a compendium of Roman laws). I would interpret "contrary to nature" differently in that context. The passage is from Book 1 Title 4 of the Digest; if we flip forward to Book 1 Title 5 we find this passage: "Slaves become our property by the Law of Nations when they are either taken from the enemy, or are born of our female slaves." This Law of Nations is as opposed to the Law of Nature-- to give you a sense of what they meant by that, a later passage reads "The Law of Nature is that a child born out of lawful matrimony follows the mother, unless a special law provides otherwise." Unless a special law provides otherwise: the Law of Nature can be contradicted by the Law of Nations, as is the case with Roman slavery.

The idea that slavery was "unnatural" in the sense of "an evil thing that should not exist" is not widely reflected in Roman philosophical writings, Roman literature, or Roman everyday behavior. I go into more depth in the answer I linked above, but inasmuch as Romans perceived anything wrong with slavery, they thought the problem was "cruel masters" and that the problem would be fixed if enslavers were just nicer to their slaves.

Whether slavery was part of the Law of Nature or the Law of Nations was hotly debated in the ancient world. The Greeks leaned towards Nature, believing that slavery was part of the natural order of the world. Many of them, most famously Aristotle, believed there existed inherently slavish people who were destined to serve superior people. In Politics Aristotle writes "The lower sort are by nature slaves, and it is better for them as for all inferiors that they should be under the rule of a master... Nature would like to distinguish between the bodies of freemen and slaves, making the one strong for servile labor, the other upright, and although useless for such services, useful for political life in the arts both of war and peace... some men are by nature free, and others slaves, and that for these latter slavery is both expedient and right."

The Romans, on the other hand, mostly believed that slavery was an "unnatural" human institution and that "slavishness" was developed in slavery (i.e. by nurture rather than nature). Seneca the Elder puts it simply: "No one is naturally free or slave. These are titles imposed later on individuals by fortune." The belief that slaves were simply normal people who had gotten unlucky fits with the fact that Romans granted automatic citizenship to manumitted slaves. There was nothing in their natures that prevented them from becoming Romans; the second-class citizenship of freedmen was connected to the learned taint of slavery and was mostly washed away by the second generation.

So basically I think Romans perceived slavery as "unnatural" in the same way that, say, aqueducts are unnatural: they're the work of humanity and not of nature.