OK,
Even though there will be problems if a user installs a newer version of your app on top on an older version, there is nothing that you can do to prevent this from physically happening (if the different versions use the same package name, the newer version will ALWAYS replace the previous version). The best that you can hope for is to post a warning in the play store product description for the app, warning the user that they should uninstall the old version before installing the new version. But again, this is not the recommended/standard way to do app updates, so I would not expect much success using this method.
Also, if the old version of the app allows the user to enter any data or set any preferences, then when the user uninstalls the old version, you are forcing the user to loose all their data and settings for the app - which would not be a good user experience.
What I would do is.....first name the new database (that comes with the new version) to be a different name from the filename of the first (older version) database. This way the older database wont be overwritten when the new version is installed on top of the old version.
Then when your app runs for the first time (after it was installed on top of the old version), have it detect if the newer database file exists, and if it does, then have your app do a one-time import of the data from the newer database into the original/older database, and then simply delete the newer database file when the import is done.
This will allow a user to install the newer version on top of an older version (which then follows standard upgrade procedures) and will prevent any file conflicts.