Sharing the information below in the event it may help. I have not tested the features as I can get by so far without them.
In the Phone app "Settings\Quick Responses", the available responses can be edited by pressing on a response. So maybe a response could be edited for the message you want to send. This might work if the person calling is using a mobile phone that can receive a text.
Also in the Phone app "Settings\Accessibility\TTY mode" there are options to turn on various TTY modes. This link has more information about using real-time text with calls
https://support.google.com/accessibility/android/answer/9042284?hl=en
In the Phone app "Sounds & vibration\Live Caption" appears to have the ability to capture speech to text during a call. This link has more information
https://support.google.com/accessibility/android/answer/9350862
Live Transcribe may be a helpful feature in general beyond voice calls
https://support.google.com/accessibility/android/answer/9158064
Beyond the Pixel phone app features, if hospital privacy policy allows, someone could text you. Similarly if the hospital uses an online secure health records system, then they could leave a message. For one of my health care providers, they use an app. After I setup an account I can communicate with the nurse and doctor, and get notified by email that a message is available. While the app is not real-time, if your hospital uses an app or an online system, then it might be another option for the hospital to provide updates.
The hospital may also have an accessibility team or staff who can communicate in the method that works best for you, and perhaps has other options beyond what is suggested above.