That Saul had some familiarity with Greek literature is shown by a number of passages, for example:
For in him we live, and move, and have our being; as certain also of
your own poets have said, For we are also his offspring. (Acts 17:28)
(this sermon, given in Athens, would have been delivered in Greek)
So there's no issue with the idea that Jesus would use Greek literature to make a point to Saul--but why quote it in another language?
Sayings can survive translation
We quote Greek literature in English all the time--even though it was originally written in Greek. There are sayings in English that were not original English compositions, but are translations. For example, "The die is cast" and "I came, I saw, I conquered" are both well-known expressions in English, even though they were both originally given in Latin.
This is not to say exact translation of idioms is helpful or common, but merely to show that an articulate idea can retain popularity in another language (this can happen with songs too, e.g. Silent Night, a German hymn, sung by people in many languages)
Whatever the exact Aramaic or Hebrew words used in Jesus' statement to Paul, the well-known Greek words of Euripides were an effective translation (Dick Harfield's post has helpfully shown the correspondence to the words of Euripides).
How to lose a job as a translator in 5 minutes
As a translator, I have learned that when translating a statement well-known in the target language, you do not free-translate the sentence. A good translator will refer to an already accepted translation. For example, if I translated "veni, vidi, vici" as "I arrived, I saw, I conquered", I would lose the confidence of my audience, because everybody knows it's supposed to be rendered "I came, I saw, I conquered."
This is why the Greek New Testament (usually) quotes the Septuagint rather than free-translating the Hebrew Tanakh. It would be natural and expected that, if Jesus told Saul something in Aramaic or Hebrew that was equivalent to a common Greek phrase, when the experience was related in Greek (whether by Saul himself or by Luke in writing Acts), the accepted Greek rendering would be used.
To return to the Silent Night example, if you compare it in German & English & Spanish, you'll find that they are not exact translations of the original German idiom, but they do convey roughly the same idea. Jesus used an idea that was known to Saul, and it was recorded in Greek in Acts using the well-known idiom.
Using the language of Saul's faith
Couldn't Jesus in heaven have just spoken to Saul in Greek? Sure. But why would he?
Greek was not the language of Saul's spiritual heritage--and this was a profoundly transformational spiritual experience for Saul. I think it more likely than not that Jesus
spoke to Saul in the language of the Jewish faith: Hebrew.
In the appendix I have supplied several lines of evidence showing that Acts 26:14 refers to Hebrew, not Aramaic.
Jesus spoke to Saul using an idea Saul was familiar with, and He spoke to Saul in the language of Saul's faith. When the idea was translated into Greek, the idiom found in the words of Euripides was the best way express what had been said.
Appendix--the trilingual world of Jesus
Although it has been commonplace in recent generations to suggest that Jesus, Peter, and others of their socio-economic status spoke only Aramaic, the evidence supporting the claim is remarkably fragile. Why have many scholars concluded that (Mishnaic) Hebrew was not spoken in 1st century Galilee? It's a claim oft-repeated but seldom argued from the evidence. Baltes offers a trenchant criticism here of the assumptions that led to this conclusion.
There is a modern myth that only extremely well-educated people speak more than one language (this myth is particularly popular among Americans--I can say that because I'm an American!). This is a sampling error. Past & present, most human beings--educated or not--have spoken more than one language. Greek did not become the lingua franca of the eastern Mediterranean until Alexander. English did not become the lingua franca of global business until the economic heyday of the British Empire.
The New Testament speaks several times of people speaking "Ἑβραΐδι". Interpreting Ἑβραΐδι as "Aramaic" worked in a scholarly world that assumed Hebrew was not spoken. The evidence of a living Hebrew language at the time of Jesus has invalidated the Aramaic interpretation.
In a region in which Aramaic, Greek, and Hebrew were spoken, "Ἑβραΐδι διαλέκτῳ" would be a plausible way to say "Hebrew", but would not be an effective way to disambiguate Aramaic. Buth & Pierce have recently argued cogently that ἑβραϊστί and related words were never used to refer to Aramaic. (see R. Buth and C. Pierce "Hebraisti in Ancient Texts: Does ἑβραϊστί Ever Mean 'Aramaic'?")
For a high-level discussion of the uses of Aramaic, Hebrew, and Greek in Jesus' milieu, see my thoughts on the Biblical Hermeneutics site here. For a much deeper dive, see this video on my channel: What languages did Jesus speak?