As people have said, your employers can't access your medical records without your permission. However, if you take more than five days then you have to get a doctor's certificate.
As Milly says, all your employer needs to know for now is the nature of your illness or injury. I can't remember whether there's a question on the employer paperwork that asks whether you saw a doctor, but if that's the case, your answer could have knock on effects in the future.
Give your employer information on question by question basis. If they don't ask for it, don't give it.