I could see it going either way.
With free access, people would be more inclined to go to the doctor for simple and small things, but in return would probably catch more serious issues early and have better access to treatment, therefor reducing the need for intensive and specialized healthcare.
Without, people avoid going to the doctor for small stuff, but end up having to go in with more complicated issues later on.
I think it would be complicated.
The demand for GPs and family doctors would increase.
The demand for hospital beds would decrease.
In the long term, things would stabilize into a new ratio of services.