I have installed the
react developer tools
The "React" tab won't show up if React can't communicate with the devtools. When the page loads, the devtools sets a global named __REACT_DEVTOOLS_GLOBAL_HOOK__, then React communicates with that hook during initialization.
(In React 0.11 and older, it was necessary to expose a global called React for the devtools to function.)
You can test this on the React website or by inspecting Facebook.
It's currently not possible to use the "React" tab in the devtools to inspect app widgets. This is due to a change in how the application scripts are evaluated in the devtools plugin; they are now run inside a Web Worker, and the plugin is unaware of this and so unable to communicate properly with React Native.
You can try http://nuclide.io/docs/platforms/react-native/#debugging__element-inspector as was suggested in issue on github, but I can't get it to work with RN 0.21