I stumbled across this question and wanted to add that the answer of a question from a link in the comments has been expanded and it is now possible to silently install the app, as long as your App is Device Owner:
https://stackoverflow.com/a/37153867/7173050
Here the part, from Android 6.0 and up:
Once your app gets the Device owner permission, we can install, uninstall and update silently without any user intervention.
public static boolean installPackage(Context context, InputStream in, String packageName) throws IOException {
    PackageInstaller packageInstaller = context.getPackageManager().getPackageInstaller();
    PackageInstaller.SessionParams params = new PackageInstaller.SessionParams(
        PackageInstaller.SessionParams.MODE_FULL_INSTALL);
    params.setAppPackageName(packageName);
    // set params
    int sessionId = packageInstaller.createSession(params);
    PackageInstaller.Session session = packageInstaller.openSession(sessionId);
    OutputStream out = session.openWrite("COSU", 0, -1);
    byte[] buffer = new byte[65536];
    int c;
    while ((c = in.read(buffer)) != -1) {
        out.write(buffer, 0, c);
    }
    session.fsync(out);
    in.close();
    out.close();
    session.commit(createIntentSender(context, sessionId));
    return true;
}
private static IntentSender createIntentSender(Context context, int sessionId) {
    PendingIntent pendingIntent = PendingIntent.getBroadcast(
             context, sessionId, new Intent(ACTION_INSTALL_COMPLETE), 0);
    return pendingIntent.getIntentSender();
}
Uninstall:
String appPackage = "com.your.app.package";
Intent intent = new Intent(getActivity(), getActivity().getClass());
PendingIntent sender = PendingIntent.getActivity(getActivity(), 0, intent, 0);
PackageInstaller mPackageInstaller = getActivity().getPackageManager().getPackageInstaller();
mPackageInstaller.uninstall(appPackage, sender.getIntentSender());
And last but not least the link:
https://github.com/android/platform_frameworks_base/blob/marshmallow-release/services/core/java/com/android/server/pm/PackageInstallerSession.java