It seems like something that some random bureaucrat made up one day when they listened to a really bad, unprofessional example of what a voicemail recording said.
Pretty soon it caught on and somebody put it into regulation. Before long, nobody knows why, but every medical or quasi-medical voicemail recording has it in their message.
Call to pay your bill at the orthodontist and the A/R lady is out of the office at lunch? She tells you to hang up and call 911 if you're having a medical emergency.
Call your pharmacy to check on your prescription strength athlete's foot cream? "Hang up and call 911 if you're having a medical emergency."
Call to schedule your 6 months dental check-up? "Hang up and call 911 if you're having a medical emergency."
Guess what! I wouldn't call my kid's orthodontist, my pharmacy, or my dentist if I was having a medical emergency!