If you do not like Gmail Messages appearing on iPhone, you can either Sign-out of Gmail or remove your Gmail Account from iPhone Mail App.
Remove Gmail Account from iPhone
Most users add their Gmail Account to iPhone as an Apple ID and later on Add the same Gmail Account to the Mail App.
Adding Gmail Account to iPhone, basically allows you to access all your Gmail Messages in the Mail App and also allows you to send Emails from iPhone using your Gmail Account.
At any time, you can prevent Gmail Messages from appearing on iPhone by toggling OFF the Mail option in Gmail section of the Mail App (Method#1 below).
Also, it easy to remove your Gmail Account from iPhone, if you are no longer using this Gmail Account or for any other reason (Method#2 below).
1. Prevent Gmail Messages from Appearing on iPhone
The simplest way to prevent Gmail Messages from appearing on iPhone is to simply logout of your Gmail Account.
1. Go to Settings > Passwords & Accounts and tap on your Gmail Account.
2. On the next screen, toggle OFF the option for Mail.
After this, you will no longer find your Gmail Messages appearing in Inbox of the Mail App on iPhone.
As mentioned above, your Gmail Account will still remain linked to the Mail App and you can start receiving Gmail Messages on iPhone at any time by toggling on the Mail option.
2. Remove Gmail Account from iPhone
If you prefer using the dedicated Gmail App or no longer want to access Gmail from iPhone, you can Remove your Gmail Account from iPhone Mail App.
1. Go to Settings > Passwords & Accounts and tap on your Gmail Account.
2. On the next screen, select the Delete Account option.
This will totally remove your Gmail Account from iPhone and you won’t be able to access your Gmail Account from within the Mail App.