You do NOT need to install a third party app.

Both Google Chrome and Firefox support this functionality. (ok, technically these are third party apps, but lets assume you are not using safari.)

Chrome:

1. Open gmail in chrome.
2. In the address bar next to the "bookmark star" you should see a grey diamond. Click on the diamond
3. Select "use gmail".
4. If you do not see the gray diamond, follow the instructions below.

If you made a mistake and selected "No" and you really wanted to select "Use Gmail"

1. With Chrome open, copy and paste this into your address bar at the top: chrome://settings/handlers
2. Under the Ignored protocol handlers section, Click on X to the right to delete.
3. Close and open Gmail.
4. Answer the question correctly this time. :)

Source: http://productforums.google.com/forum/#!topic/chrome/hnRteY7bHfE

Alternative way, if you need to do this programmatically.

Firefox:

See @Am1rr3zA answer on this question. Click here to jump to his answer.

Safari:

See @fpotter's answer on this question. Click here to jump to his answer.

This is not possible on iOS 13 and lower. (Or iPadOS 13 or lower)

In order to utilize email functionalities from within apps, you will have to enable the Apple Mail app. I recommend turning off all notifications for Mail if you want to use the Gmail app as your primary iPhone mail client.