The first time your app is installed, install a separate application/package ("watcher").
The only classes "watcher" contains is a BroadcastReceiver that listens for ACTION_PACKAGE_REMOVED
Add a BroadcastReceiver to your application that also listens for ACTION_PACKAGE_REMOVED
When a intent is broadcast to one of your receivers, check if the other component is still installed. If is isn't (the user just uninstalled it), prompt for the password - if it's wrong, reinstall the other component. If it's right, uninstall yourself.