the work performed by a doctor would not be prohibited because that is not the type of work that is prohibited. The type of work that is prohibited is "melachah." Which is, for example, the type of work that G-d would perform. For instance if a doctor cured a person a disease on the Shabbat it would be acceptable because his is not the work of G-d. If a Rabbi cured the same person of the same disease on the Shabbat it would be a violation because his is the work of G-d. The reason that it was a violation for Jesus to heal the sick on Shabbat is that he was not performing the type of work that a doctor would but the type G-d would.
This would also explain why the Jewish leaders were upset with what Jesus had done but not surprised by it. They realized that he was acting under divine power and not as a doctor. Why do you think the religious leaders would be surprised by the healing power of G-d?
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