Can you check that you have the latest iOS version installed? If you perform an update, then try as well to see if you can update the Stadia app after this as well.
Hopefully that solves the issue, otherwise maybe try a simple restart of your device and then try again.
If that still doesn't solve the issue, it might help if you get in contact with the Stadia support, maybe they have some more ideas on what to try in such a case.
