Greek and Babylonian astronomers definitely did not predict the eclipse of May 585 BCE.

Without knowing Carlin's exact phrasing, I presume that either he or his source said that the Ionian Greek thinker Thales may have drawn on Babylonian records to formulate a prediction. This is a hypothesis made by G. S. Kirk in The presocratic philosophers (Cambridge, 1960), p. 80, but there is no evidence for it; Oscar Neugebauer has shown that there is no good evidence of such records; and even if the records did exist, they would have to have recorded solar eclipses that weren't visible in Mesopotamia. For further details I recommend Alden Mosshammer's 1981 article 'Thales' eclipse' (JSTOR link).

The only claim of 'prediction' that we have comes from Herodotos 1.74, writing about 160 years after the event.

A war went on between the Lydians and the Medes for five years. The Medes often beat the Lydians, the Lydians often beat the Medes. There was even a night battle, of a sort. They were evenly matched in the war, and when there was an encounter in the sixth year, during the battle it happened that day suddenly became night.

Thales of Miletus had advised the Ionians in advance that this transformation would happen, setting this year as a boundary, in which the change did in fact take place.

There are several other ancient reports of the incident, but they are all derived from Herodotos and therefore add no further information. The ultimate source for them all is Herodotos' wording.

So, let's make sure to observe carefully what Herodotos actually says.

###1. The eclipse

Herodotos reports that 'day suddenly became night' (τὴν ἡμέρην ἐξαπίνης νύκτα γενέσθαι), and he calls the incident a 'transformation' (μεταλλαγή).

He uses 'day became night' when describing the same battle elsewhere too (1.103). But this phrasing is not straightforward. He doesn't say 'the sun was eclipsed', which is the phrasing that Thucydides uses when talking about solar eclipses. He also doesn't use the phrasing 'the sun was darkened in the sky', which he uses in connection with a separate incident (9.10). But he does use the 'day became night' phrasing again in connection with an event that supposedly took place in 480 BCE (Herodotus 7.37) -- but there was definitely no solar eclipse in Greece in 480 BCE.

So Herodotos' phrasing does not strongly point to a solar eclipse.

###2. The prediction

Herodotos' phrasing --

Thales of Miletus had advised the Ionians in advance that this transformation would happen, setting this year as a boundary, in which the change did in fact take place.

-- does not sound much like someone predicting a particular day. So Herodotos' phrasing does not point to a prediction either.

###3. What would it have taken to make such a prediction, anyway?

The usual explanation for how someone might have predicted a solar eclipse at that date tends to revolve around saros cycles (a period of 223 lunar months) and/or exeligmos cycles (3 saroi). Saros cycles are only useful for predicting lunar eclipses: when solar eclipses repeat after one saros, it's at a different place on the earth's surface, so this fact would have been useless to an ancient astronomer. And an exeligmos cycle couldn't have been used to predict the 585 BCE solar eclipse, as the 585 eclipse was the first one in its exeligmos series to reach as far as the Mediterranean. Also, the earliest known Greek use of the exeligmos dates to about half a millennium after this battle.

Predicting solar eclipses correctly requires a very accurate (and heliocentric) model of the relative movements of the earth, sun, and moon. There are workarounds that don't require heliocentrism, but the earliest effective workarounds only began to appear in 4th cent. BCE Babylonia and 3rd cent. BCE China (Steele 1997, 1998) -- several centuries later, and thousands of kilometres away from Thales' supposed prediction.

As Mosshammer puts it,

There is little value in having a precise, astronomically calculated date, time, and place for a battle that did not happen.

Some further reading: Couprie 2004 demolishes the central theory, and tries to come up with an alternate explanation of why Herodotos wrote what he wrote; Querejeta 2011 is sceptical even of that.