There is a large number of private laboratories/clinics in Greece that do COVID-19 diagnostic testing on short notice and can provide a certificate with your results in English.
Regardless of the which country you are heading to the most important thing to remember is that the type of COVID-19 diagnostic test required in all cases is the PCR (Polymerase Chain Reaction) type of test, also known as, molecular test, nucleic acid amplification test (NAAT), RT-PCR test, or LAMP test. To be on the safe side though, make sure the certificate you receive refers to it as "PCR" or "RT-PCR" which is the name everyone will recognise and accept as valid.
Then, depending on which country you heading to, things can become more demanding. For instance, the airline employee at check-in or the immigration officials at the port of entry may require that you produce documentation or online evidence that the laboratory you did the testing at is officially certified/recognised for this type of testing by the Greek government. Given that the Greek government's sites are usually in Greek and pretty terse in quantity and quality of easily accessible information this could prove hard to do. I did do some research and thankfully there is official info online, in English, that can be used for that purpose.
The official Greek certification body for laboratories (including medical) has a minimal English version of its site here: esyd.gr
The site is the typical buggy, incomplete, non user friendly Greek public sector mess. It may take some time but eventually you will hit this page here (screenshot below) which allows you to get a listing of all medical labs certified to ISO 15189 standard.
The results are not necessarily labs that do COVID-19 PCR diagnostic testing and they are a mix of public and private entities. Using this listing you can find those labs that are listed under ISO 15189 certification for "human samples", like the one below, and which, in their "Scope of Accreditation" document, linked into their "Accreditation Details" page, list COVID-19 PCR testing as well.
If you already know of a lab that does COVID-19 PCR testing and want to check if it is officially certified then you can use this search form here.
Finally, for those able to read Greek there is a precompiled list of certified COVID-19 PCR diagnostic testing labs here.