Do you know what this medal is?

I would start digging in Chaplain O'Neill's past to see if he had a connection to the Equestrian Order Of The Holy Sepulchre Of Jerusalem (EOHJC).

The medal very much looks like a combination of the EOHSJ Jerusalem Cross with the smaller crosses (representing Matthew, Mark, Luke and John) replaced with the four medals (Sacred heart, St. Christopher, St. Joseph, and Miraculous) from a Catholic four-way cross.

Those was (supposedly) the last words of the last pagan Emperor of Rome Julian; son of Emperor Flavius Julius Constantius, half brother of the Great Constantine. It's basically interpreted as referring to "GALILAIE" Galilean as Christ and the Christian Relgion overcoming Julian's persecution of Christians during his reign. I believe the vertical lettering is some type of number...
Further reading about the Greek lettering it maybe related to Engineering or Masonic organizations..that's as far as I researched. My eyes are a little fried from the detailed research that I started around 7pm EST.
Let me know what info you will further dig up

https://en.wikipedia.org/wiki/Against_the_Galileans

https://en.wikipedia.org/wiki/Julian_(emperor)


https://www.newadvent.org/cathen/08558b.htm

https://www.cambridge.org/core/journals/journal-of-roman-studies/article/abs/death-of-julian-the-apostate-in-a-christian-legend/2D142F4F8AE753B3DAF3C732FE6176EB

https://en.wikipedia.org/wiki/Julian%27s_Persian_expedition

I would guess that it's a Greek Orthodox religious award. The cross is Greek as is the alphabet being used. Inasmuch as the four images appear to be religious icons, I would expect that the purpose of the medal is religious (as opposed to anything martial- or Olympics related. I looked through the first seven pages of "greek medals" on eBay and didn't find anything like it. You may want to continue that search. There are more pages.