React Native's ecosystem is so full of such magic that we might forget were we started.
Before all those fantastic tools we had to deal with manual App Store submissions (and Google Play for that matter).

Solution of choice for mobile Apps deploy automation (and maybe behind the curtain of the tools above... who knows) is fastlane.
If you want to dig deeper, or if you need a different solution tailored for your needs here are Daniel Banck and Daniel Schmidt talking about it.

Daniel Banck is also keeping pace with React Native udpdates and we can expect him to extend the guide from App Store to Google Play as well.

Deploying a React Native App with Fastlane blogpost by Daniel Banck

React Native deployment with Fastlane blogpost by Daniel Schmidt