The health industry is under strickt legal obligation to protect your medical information. Rather than open themselves to being sued or fined, they probably won’t let you use the app on an open device that anyone can pick up and use.
There is a good chance the service your doctor is using has a website with all the features of the app. My clinic (and many others around here) use MyChart. it has both an app and a secure website. They work much the same. If they do, you can probably use the site instead of the app.