Specifically, what criteria in Sparta was used to determined if a newborn was fit to become a warrior or thrown off Mt. killbaby?

by The2500
Iphikrates

We're skipping past the real question here: did the Spartans actually throw imperfect newborns off a cliff? Probably not. I've answered this in much more detail here.

As I point out in that post, the entirety of our evidence for this Spartan practice is a single passage in a very late source affected by several centuries of deliberately misleading Spartan propaganda. It does not give any details about the selection process, so we can't even answer the question how Sparta would hypothetically judge babies. Other sources advocating eugenic practices are similarly vague. Take Aristotle for example, who argued that the ideal state should expose babies deemed defective (but tellingly does not mention Sparta as a place where this was already done, reinforcing the point that Sparta did not actually do this):

As to exposing or rearing the children born, let there be a law that no deformed child shall be reared.

-- Aristotle, Politics 1335b.19-21

That is the full extent of his comment on the matter, except to add that some states had laws preventing the killing of newborn babies, in which case Aristotle recommends fertility restrictions enforced through mandatory abortion.

The word he uses to define the babies he sentences to death is πεπηρωμένον, "being defective/maimed/incapacitated". Since it would be impossible to know what cognitive problems a newborn baby may develop, we have to assume that he is referring to obvious, visible physical deformities only.