flutter - How to keep user signed-in with Google
Users already stay authenticated. After you restart the app, Firebase reads the credentials from disk, and refreshes the user's token. Since this requires a roundtrip to the server, it happens asynchronously. So be sure to await _auth.currentUser() to get notified of the user's status.
Whenever I'm wondering how to do such things, I look at the FlutterFire sample app. This specific line can be found here.
Yes you need to do the auth.currentUser() function in order to keep users authenticated and the best part is that this function will work even if the user is offline, which makes it very versatile.