The answer is - no, you can't do that - not at least the Flutter* apps. As @Abion47 said, you can publish the new APK and ask users to install it. If you app involves back-end then you can check the (client) mobile app version and display the alert if the version is below minimum version required.
On the iOS side, it is slightly more difficult. For internal use, the organisation needs to sign-up with Apple's enterprise program. You can then distribute the app internally.
*- I've seen certain apps downloads the zip (or whatever) file, and updates themselves without actually going through App update process via AppStore/Playstore. However, it is more of a web part within the particular app which gets updated. The app version as such remains same.