Is it possible to initiate a fully automated app installation (after a user clicked a "Install app" button)? Or do I simply have to open the respective Google Play page in the browser, and the user has to handle installation from Google Play as usual?
No. It is impossible to install apps from the API without user intervention. There are a number of reasons for this, you could perform privilege escalation, to mention one. Instead, you just have to redirect the user to the Google Play app.