What does “medical scare” mean?

A medical scare can be anything where your health was suddenly in danger. It's when you thought that there could be something seriously wrong - could be something like an emergency, but could also be a serious illness. Often the idea behind it is that in the end, nothing major was really wrong. If you say, "I had a medical scare last year," the idea is that you felt ill and were investigated for medical problems.

A more common expression (see Ngram) is health scare that can refer both to a public and a private, single medical episode.

Both health and medical scare refer to:

• A situation characterized by alarm or anxiety about the risk of developing or being diagnosed with a particular illness or condition:
  • he caused an international health scare by travelling with a dangerous form of tuberculosis
  • she’s back in good spirits again after a recent health scare

(ODO)

From CNN.Com Fed Chair Yellen is 'fine' after brief health scare

I think the phrase means rather that there was a concern about one's health arising from a medical assessment or test, rather than a personal fear about a symptom. For instance, if a blood test turns up something that warrants a more extensive test, such as an x-ray or scan, to rule out, say, cancer, that would be a medical scare. It's a medical scare because it arises from interaction with the medical profession, as opposed to a health scare.

The period of not knowing whether you have cancer, from the blood test results to the scan, would be the medical scare. It's a scare because it frightened you, but it didn't turn out to be an actual health problem, which would have been very serious. Having dizziness or feeling ill wouldn't be a medical scare; that I think would be more like hypochondria, if you thought it was a portent of a more serious condition.