How to stop Gmail app checking emails in background?
As the GMail app is a third party app not built by Apple, it is not allowed to run in the background to check for mails. So closing the app stops it from checking for mails.
However, it probably will still get notifications about new mails pushed from Google's server. But this notification is not initiated from your phone.

You can switch off notifications for the gmail app in the iPhones settings.app under notifications.