Joann, what kind of phone is it exactly? Make and model? How new is it? It's possible it doesn't have all of the most common LTE frequencies used in Greece. It may have worked in previous years on 3G (is it the same phone?) but many European mobile companies have shut down their 3G networks.
You might go into Settings and try to manually have your phone search for a mobile network. The way you do this is different on every phone. In my Moto X4 Android phone, for example, I go into Settings -> Network & Internet -> Mobile Network. From there, I first make sure Roaming is turned on. There's an "Advanced" menu which lets me choose a "Preferred network type." Mine is set to LTE. If yours says "CDMA" change that to "LTE" (change it BACK when you get home).
Mine also has a setting in this menu, "Automatically select network," which is ON for me - usually this is what you want. But I can turn that off and then choose a network instead. Maybe your phone can see a network that you would be able to choose to connect to.
You won't see any "Verizon" networks anywhere in Europe, because Verizon doesn't have service there. Instead, they have roaming partners.
One option is to buy a Greek SIM card for your phone, assuming the problem is Verizon's roaming not your phone, so at least you can use it while in Greece. (Go into a mobile store in Greece.) If you have to do that, then scream at Verizon when you get home and get them to reverse the $100 international roaming fee you paid, at least. Your phone will have a Greek phone number while the Greek SIM is in place.
Here's more info about Greek prepaid SIM cards, the different companies, etc. There seem to be three main companies operating there: Cosmote, Vodafone, and Wind. (Verizon must have at least one of them as a roaming partner.)