We have nationalized health care here in Greece, of course there are private practice doctors and private hospitals, and whoever wants to, can go to them.
If you have an emergency, regardless if it is a serious issue like a heart attack or you simply closed the car door on your finger, you can visit an emergency room in a public hospital, wait your turn (or be rushed in) and be treated on the spot, or hospitalized. Doesn't matter if you are Greek or any other nationality, or a tourist. An ambulance will come and pick you up if you are unable to move yourself.
Of course do not expect what you see in the "ER" in terms of caring professionals, the sheer volume of outpatients exceeds the physical and mental capacity of the doctors and nurses, but usually they do what they can, and more. Couple that with a "Oh my God, I was shaving in the morning and cut my cheek a little, I am in the verge of death" national mentality, and the persons going to the hospital double. Add their close relatives who try to come with them in case the "patient" needs any "assistance", and this becomes the grim picture of Greek public hospitals. and of course everyone tries to get first in the waiting line...
Interestingly, I have no problem with that... maybe I have more patience than the average Greek, but I think I have been treated well in the past, and hope so with the future. I had a vocal cord nodue removed, my father had a pacemaker installed and was treated for prostate something, my mother was hospitalised for cardiac arrythmia (minor), etc, all in a public hospital, and for FREE. (Not exactly free, because every working person or pensioner gives a percentage to the Health fund).
The system of course is not without problems, this being Greece and not a civilized country. Too many to mention. Hoping for the future...