The platform is fine, however the custom software you create to allow patients to add appointments and such may or may not be, depending on how it is written.
I know medical people have to adhere to stringent regulations, such as HIPPA. Several of the standards required by such laws would not necessarily be included in WordPress itself, although the custom software built on top of it could implement them. Encryption of patient information, for example.
WordPress as a platform can be customized to do many things. However fundamentally it is a semi-generic Content Management System. It’s made to create web pages, not to manage secure data. It would be a fine platform to build the web interface on, but the actual data storage and transmission of the “private” data should be handled by some other certified system.
For example, you wouldn’t want to store patient records inside the WordPress database, but it would be perfectly possible for a WordPress add-on to communicate with an external system to manipulate those records as necessary.
You also need to consider what information is legitimately “private”. Appointment timing, for example, may not be. In which case you could store it anywhere.