The guys at Auth0 are doing a pretty good job pursuing their goal that they share as their motto "Eliminate the friction of identity for your applications and APIs".
Luckly, for us, as developers they got interested in React Native and shared their tutorial and component in order to use TouchID for Authentication in React Native apps.

We’ve checked React Native components and couldn’t find one that allows us to use TouchID, so we went back to good ol’ Objective-C since there is no support for Swift yet. We decided to code a React Native Module that would allow everyone to use TouchID. Here in Auth0, we've already implemented a TouchID authentication flow for Lock. So we decided to reuse that library and wrap it in a React Native compatible library. In other words, we created Lock.ReactNative.

The component is on GitHub accompanied with an example app.