Well yes, I think you're right that all immigrants who entered USA via New York went through Ellis Island (during the years Ellis Island was open).
It is wonderful that you have the documents you posted! Thanks for sharing them. Here are:
1. doctor exam papers? 2. another health document 3. naturalization card from many years later in the USA
I don't see the passenger manifest of the ship however. It is a list of passengers who are getting off the ship in New York. Ancestry.com website (pay site) has a digital collection of the Ellis Island passenger manifests, and so does the Ellis Island website (free).
Have you ever tried searching for your grandfather's Ellis Island record - the ship passenger manifest, showing he was on that ship when it arrive in New York?
I have many times been unable to find something that another person later finds. You very well may have better luck than me, if you try there.
Sometimes the passenger manifest tells who the passenger is going to join in the USA. It's possible the passenger manifest might tell that and/or other useful clues. No guarantees.
The doctor's report seems to be made at the start of the voyage, from what you said before. Ellis Island also had a medical examination at the end of the voyage, that passengers must pass in order to enter the country. If ill, a passenger stayed on the island in quarantine until recovered, so that no communicable diseases were brought into New York by sick passengers.
I suppose it would be useful to examine a passenger at the start of the journey - so that the boat captain would not be charged with the care of someone not allowed into the USA at all (for health reasons).
Also, read about naturalization papers. I think you might find another (although perhaps no additional information, I don't know). Should be at least two for every immigrant: declaration of intent to become naturalized, and final naturalization. You have posted the latter.
Petros /Peter seems to have surname spelled without the J in the papers you posted ... do I see that right? I did not search at all without the J. So there's an example of a clue that comes forward when we look closely at the actual documents.
Thanks again for all this.
Happy Valentine's Day (belated) to you too!