Medical ID - unable to change

According to this page in the iPhone User Guide, http://help.apple.com/iphone/11/#/iphec639eb15 there does not seem to be anything special that has to bed one to make changes. If you have restarted the phone, then the next step is to force restart.

• On an iPhone 8 or iPhone 8 Plus: Press and quickly release the Volume Up button. Then press and quickly release the Volume Down button. Finally, press and hold the Side button until you see the Apple logo.
• On an iPhone 7 or iPhone 7 Plus: Press and hold both the Side and Volume Down buttons for at least 10 seconds, until you see the Apple logo.
• On an iPhone 6s and earlier, iPad, or iPod touch: Press and hold both the Home and the Top (or Side) buttons for at least 10 seconds, until you see the Apple logo.

Do you have any restrictions turned on? Check Settings>General>Restrictions to see. This is the only support document I can find right now about Medical ID. Set up your Medical ID in the Health app on your iPhone - Apple Support

Where are you attempting to change the Medical ID information? You would go to the Health App, tap on the Medical ID on the bottom right of the screen, then in the upper right you should see Edit.

I'm not sure where you are talking about the ICE contact, unless you are referring to the Emergency Contact List in the Medical ID area. You need to explain a little better what the problem is and where you are working.

I go to the Health App. I tap on the Medical ID icon in the lower right corner. That opens Medical ID. I go to Edit in the upper right corner. I can see the information entered, and to the left of them, there is a ➖ to be able to delete information. Scrolling down to the bottom of the screen, I see a ➕ to be able to add an emergency contact. And, if I scroll further, I can see a bar that reads, Delete Medical ID.

If you do not see that, then it is very possible that you have a restriction on about changing account information. If you still do not see that, or are unable to do anything, I suggest you make a Genius Bar appointment at the nearest Apple Store and seek help.

I did not mean just for this app, I'm speaking overall restriction on changing account information. Other than that, I cannot really recommend anything else. My Medical ID is rather involved, and I would rather not delete any information from it, nor delete the entire ID to test for you. I believe at this point that Apple Support is your best bet.