Doctors spend at least 8 years of their lives training for their profession. By the time they become a full fledged physician, they've absorbed a breadth of knowledge about nearly every aliment man can have. Still, they can't predict everything. Medical records are needed constantly, because they help narrow down the possibilities of illnesses one might have. There is no reason to not let docors see the records, because they are sworn by oath not to reveal them to anyone else.
The EU should allow EU citizens to go to any hospital within the EU and allow the treating physician to have immediate and instant access to the patients medical records. This makes it easy for citizens to go where they want and who they far comfortable with seeing. No citizen would argue this.
Having centralized medical records, so long as privacy is maintained, is a great innovation that can help someone get better medical care. Central medical records help doctors make sure someone's prescriptions and treatment aren't contraindicated to each other. Medical records that can be easily accessed saves lives and saves money, plain and simple.
The number one concern of any person in a hospital receiving treatment should be the effectiveness of their care and their overall health. This goal is best met by allowing an easy to use and access system of organizing patient medical records. Things like allergies and health history would be made easier, speeding up the appointment time and better individualizing care for the patient.
Its way too complicated to have one hospital system because of the huge differences in the quality of care in countries, and the difference each person in each country spends vs consumes. The hugely complicated process of having a European health care system would leave to gross abuse and inefficiency.