medical certificate; doctor's note
A formal document issued by a doctor or medical institution stating a diagnosis, condition, or fitness for work/school.
I got a medical certificate from the hospital to take time off work.
Please submit a doctor's note to the school.
処方箋 is the slip used to obtain medication, whereas 診断書 is a doctor's written certificate stating a diagnosis for administrative or legal purposes.
Compound of 診断 (diagnosis) + 書 (document).