To begin, it is important to note that there is something of a shift over a 1400 year history of ancient Greece, between say 1600 BC and 200 BC. The Mycenaeans, for example, typically only took women slaves as spoils of war, though there is plenty written to refer to male slaves as well. These were typically prisoners of war, and that tradition would continue well into the 1st millenium BC, but also come to encompass chattel slavery and serfdom as well. During the eary 1st millenium during the time of Homer, the prevailing view saw slavery as the normal outcome of warfare, that a people who lose should accept their lot as slaves.

Chattel slavery, or owned bondage, did exist in the classical period, of course. Though still most slaves were more comparable to serfs being tied to the land. The Messenians (Helots) of Lacedaemon are the most notable example of this though it existed in many if not all city states.

I think this question is best answered by Thucydides. While the comment is not related to slavery per se, it is telling of how the Hellenic states viewed broader culture and society and may give you insight.

In History of the Peloponnesian War, Thucydides describes a Peloponnesian League meeting between Sparta, Corinth and the others where Corinth lays out their case for attacking Athens preemptively following the Corinthian defeat at Sybota by a combined Athenian and Corcyran fleet (and the Athenian demands to Potidaea). An Athenian delegation happens to be in Lacedaemon during the meeting and enters the congress to speak in defense of Athenian actions, in which they say "...for it has always been the law that the weak should be subject to the strong." Ignoring all other context of that statement, it is very telling about how the ancient Greeks would have viewed such matters.

Still, this view is contextualized by the Athenian practice of greeting newly acquired slaves with fruits and nuts, a ritual also used to greet a newly wedded wife into the home. Slaves were generally invited to participate in household worship and ceremonies as well. Xenophon on the other hand wrote that slaves "should be treated as domestic animals" and goes on to suggest that it is important to punish harshly the bad, and reward generously good behavior by slaves. This is important, as it shows an attempt to dehumanize slaves, which could be interpreted as a need for justification. Most written works dealing with Utopian idealism take slavery as a given, such as in Plato's Republic. Aristotle was particularly adamant about the need for slavery, going so far as to argue that a non-slaving society simply would not function, comparing them to essential tools. I think in this way it shows that while the majority of Greeks saw little need to justify their ownership of slaves, philosophers and perhaps other social elites actively tried to dehumanize the slaves to make ownership more palatable to the ostensibly civilized Hellenic people. Cultural practice suggests that they viewed them as human rather than simply tools or property. Most likely saw slavery as a matter of course with little need to justify the act itself, just the fair treatment of the slave.

Edit: I hope this doesn't come off as rambling. I originally wrote a much longer version but trimmed it up so it may sound disjointed in some areas. I wasn't sure how much context you needed but didn't want to come off like I was insulting your intelligence. Still, I think this should adequately answer your question on the matter.