No, that is just not going to work well.
If you leave home on June 6 and return home on June 20, that means you will land in Greece on June 7 after a long, quite possibly sleepless, overnight flight. You may be barely functioning on June 7. Most of us don't count our arrival day as a real sightseeing day. We walk around outside, trying to stay awake, and then we crash early and hope to be semi-normal on Day 2. June 20 will be devoted to getting yourself to the airport and flying home. So your real, fully usable, vacation-in-Europe days are June 8 - June 19. That's just 12 days.
Unfortunately, there are no super-speedy trains to whisk you around Greece and Turkey as you would find in some western European countries. Getting out to and between the islands requires dealing with airport or ferries. On the mainland you'll probably be depending entirely on buses. ViaMichelin.com estimates the Athens-Meteora driving time (in your own car--not on a public bus making who knows how many stops) at over 4-1/4 hours one-way.
It will be a slog every time you move to a new hotel, reducing what started as a full day to just a few hours of sightseeing time. To maximum your real relaxation time, you need to minimize the number of times you change hotels.
I'll leave it to others to suggest the maximum you should attempt in the time you have, but I think Turkey is a no-go unless you want to give up most of Greece. We have some people on the forum who travel often to Greece, and they can give you a realistic idea of how much can be covered in the time you have.
One thing to keep in mind is that if your flight home is from Athens, it would be prudent to hit Athens at the end of the trip. Otherwise, you will be at risk of missing the return flight if you get stuck on your last island stop because your flight or ferry is canceled.
