A learned classical Greek would probably have given you Aristotle's answer.

Cited from Politics, lib. 1, v:

"But is there any one thus intended by nature to be a slave, and for whom such a condition is expedient and right, or rather is not all slavery a violation of nature?

There is no difficulty in answering this question, on grounds both of reason and of fact. For that some should rule and others be ruled is a thing not only necessary, but expedient; from the hour of their birth, some are marked out for subjection, others for rule.

And there are many kinds both of rulers and subjects (and that rule is the better which is exercised over better subjects- for example, to rule over men is better than to rule over wild beasts; for the work is better which is executed by better workmen, and where one man rules and another is ruled, they may be said to have a work); for in all things which form a composite whole and which are made up of parts, whether continuous or discrete, a distinction between the ruling and the subject element comes to fight. Such a duality exists in living creatures, but not in them only; it originates in the constitution of the universe; even in things which have no life there is a ruling principle, as in a musical mode. But we are wandering from the subject. We will therefore restrict ourselves to the living creature, which, in the first place, consists of soul and body: and of these two, the one is by nature the ruler, and the other the subject. But then we must look for the intentions of nature in things which retain their nature, and not in things which are corrupted. And therefore we must study the man who is in the most perfect state both of body and soul, for in him we shall see the true relation of the two; although in bad or corrupted natures the body will often appear to rule over the soul, because they are in an evil and unnatural condition. At all events we may firstly observe in living creatures both a despotical and a constitutional rule; for the soul rules the body with a despotical rule, whereas the intellect rules the appetites with a constitutional and royal rule. And it is clear that the rule of the soul over the body, and of the mind and the rational element over the passionate, is natural and expedient; whereas the equality of the two or the rule of the inferior is always hurtful. The same holds good of animals in relation to men; for tame animals have a better nature than wild, and all tame animals are better off when they are ruled by man; for then they are preserved. Again, the male is by nature superior, and the female inferior; and the one rules, and the other is ruled; this principle, of necessity, extends to all mankind.

Where then there is such a difference as that between soul and body, or between men and animals (as in the case of those whose business is to use their body, and who can do nothing better), 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. For he who can be, and therefore is, another's and he who participates in rational principle enough to apprehend, but not to have, such a principle, is a slave by nature. Whereas the lower animals cannot even apprehend a principle; they obey their instincts. And indeed the use made of slaves and of tame animals is not very different; for both with their bodies minister to the needs of life. 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. But the opposite often happens- that some have the souls and others have the bodies of freemen. And doubtless if men differed from one another in the mere forms of their bodies as much as the statues of the Gods do from men, all would acknowledge that the inferior class should be slaves of the superior. And if this is true of the body, how much more just that a similar distinction should exist in the soul? but the beauty of the body is seen, whereas the beauty of the soul is not seen. It is clear, then, that some men are by nature free, and others slaves, and that for these latter slavery is both expedient and right."

Additional details in lib. iii and iv, which I posted earlier but redacted, as it was less on topic than this section.

Roman jurists like Gaius also had intriguing thoughts on nature vs slavery. I'm happy to answer questions.

I can't speak directly on the topic of slavery, nor speak for all of ancient Greece, but in the time of the empire, Athenians had a sort "might makes right" mentality about invading and enslaving cities. Thucydides records this attitudes in the Melian Dialogue Here:

Athenians. When you speak of the favour of the gods, we may as fairly hope for that as yourselves; neither our pretensions nor our conduct being in any way contrary to what men believe of the gods, or practice among themselves. Of the gods we believe, and of men we know, that by a necessary law of their nature they rule wherever they can. And it is not as if we were the first to make this law, or to act upon it when made: we found it existing before us, and shall leave it to exist for ever after us; all we do is to make use of it, knowing that you and everybody else, having the same power as we have, would do the same as we do. Thus, as far as the gods are concerned, we have no fear and no reason to fear that we shall be at a disadvantage.

They do, in fact, end up sacking a city and enslaving everyone after this argument. In a less specific way, I would also say that slavery was very different in Ancient times. Slaves were taken from war, debtors and other sources. It was not unilateral race slavery like American slavery. Nothing about a man's nature made him a slave. This particular man had been in a position resulting in slavery, such as selling himself into slavery, owing a debt or being taken as a war captive.

Seneca is a Roman, so that makes him at best marginally useful to answer your question, but he does speak extensively on slavery from the point of view of the Stoic philosophy, which is rooted in Greece in your time period. If we allow Seneca to speak for the Greek stoics (which I admit is problematic, but he addresses slavery specifically), then Stoicism teaches that masters have a duty to be humane in their treatment of slaves (so slaves are not without the rights due to humans as possessors of logos), but there is nothing necessarily inherently wrong with some people being slaves. Indeed, most people are enslaved by their lack of virtue, and only the sage is truly free; moreover, his freedom does not come from anything so base or, to a Stoic, perhaps unimportant as whether he happens to legally belong to someone else.