You can get the navigator through refs property: https://facebook.github.io/react/docs/more-about-refs.html. It's part of react (not specific to react native). It's not obvious from the react-native docs that there is a number of 'react' features that can be used in react-native, so i'd really advise to take a close look at react in general.
Note however, there is a good reason Facebook does not mention refs explicitly and loudly. Refs is really not a "go-to" way of accessing component. Your case might be of course different, but it's likely that the Google sign-up is not in-fact "external". It might actually be part of one of the components in the hierarchy tree above the navigator (in which case you can pass the state change down the tree).
Quoting from the summary of the "More about refs" document above:
If you have not programmed several apps with React, your first
inclination is usually going to be to try to use refs to "make things
happen" in your app. If this is the case, take a moment and think more
critically about where state should be owned in the component
hierarchy. Often, it becomes clear that the proper place to "own" that
state is at a higher level in the hierarchy. Placing the state there
often eliminates any desire to use refs to "make things happen" –
instead, the data flow will usually accomplish your goal.
Again - your case might be different and using refs might be perfectly justified, but if you are tempted (for example) to separate out all the Google-related stuff to separate object and if that makes the sign-up "external" - think twice. React really encourages putting all things related to a "component" logic in one place (the component) - even if that includes various technologies and external APIs.