Or, even better (at least from my point of view): touch the button at the bottom left of your S4 to open a menu, then select the gear for, "Settings". This will take you to a different screen. Then at the top of this new screen, select "Accounts". Then under "My accounts" select your gmail account and at the bottom right of your S4, select, "Remove account". Problem solved.
It was driving me crazy the amount of audio notifications my S4 was giving me, all because of that stupid gmail app on the S4. From now on, whenever I want to check gmail, then I will just use the internet on my S4 and simply go to gmail.