I want a doctor to examine my baby to see if she’s growing well.
In order to diagnose a patient, you must first go to examine a patient. When you examine them you know what's wrong with them and then you can diagnose them, which is why this verb is to examine.
The reading is the same as 見る, which you have to do when you are examining a patient. So, you look and see all you can about the patient while you examine them, making this reading み as well.