If you implemented your "business logic" in, say, Redux or some other Flux implementation then you could share a massive amount of code between the two; however because React Native acts as a fork of React, having both in the same project (same package.json) will run into issues as far as I'm aware; so you could share code, but not have a single project that compiles to both, at this point in time
EDIT: See below, this is no longer the case as of the latest versions?
As of one or two releases ago React Native no longer ships with a forked version of React, and just depends on the upstream "React" library like any other project. So it's compatible with most things in the ecosystem now that don't depend on a DOM being available.
Oh really? What version is that at, we've been upgrading at every release, but I could've sworn we still ran into issues even with npm v3 with external libraries installing React when installed, rather than relying on the upstream that should already exist. Might need to upgrade again!
Edited to add:
We're running v0.21.0 here, and I know there were still issues with certain libraries causing conflicts even running up to v0.20.0 -- we've not had any issues since upgrading mind you.
EDIT: See below, this is no longer the case as of the latest versions?