Apple - How do I remove Gmail from my Mail app without removing Gmail account from system preferences?

Simply disable your Gmail account by going into the menu bar, MailPreferencesAccounts and uncheck Enable this account under Account Information for your Gmail account.

The account will remain under System Preferences, but Mail will no longer fetch your email nor, therefore, notify you of incoming email under your Gmail account.

Enable this account

Although the answer above is valid - there is a nicer (I think) way. You don't need to delete your Google account, and you don't need to disable it specifically in Apple-Mail application.

If you open your System Preferences, and select the Internet Accounts panel, you'll see your accounts on the left, Google being one of them.

If you select your Google account on the left, you can then choose which services this account will connect to. You can uncheck "Mail" while retaining other Google Services.

Here is my system preferences screenshot System Preferences Screenshot