Removing Emergency Contact is as simple as adding them. To remove any contact information, go to the iPhone Settings, then Emergency SOS, and tap "Edit Emergency Contacts in Health". Once you are on Medical ID, select the "Edit" button (upper-right corner). Now scroll down to Emergency Contacts, tap on the red circle with the "minus" sign, and the Delete button will appear. After deleting the contact, press "Done" to save the information.