In our App we have Google sign-in integrated as explained here :
https://developers.google.com/identity/sign-in/web/sign-in
Our App is also looking for users Gmail account using serverAuthCode provided by Gmail. So can we use Google sign-in for Gmail API access or we need to have Gmail sign in as explained here :
According to Authorizing Your App with Gmail:
All requests to the Gmail API must be authorized by an authenticated user. Gmail uses the OAuth 2.0 protocol for authenticating a Google account and authorizing access to user data. You can also use Google+ Sign-in to provide a "sign-in with Google" authentication method for your app.
If you watch the video from the Google Sign-in link you provided, you will know that this tool is used to handle "signing-in". Now if you want to use Gmail APIs, you'll have to do more. Aside from OAuth, you will need to handle authentication and authorization, as well as scopes.
Since you seem to be working on Android and iOS, check this quickstarts. They have sign-in for Gmail API implemented:
