Ah, so you have an app that links to the "fiddle" and downloads the RN JS bundle and runs it for you, instead of an emulator. Nice!
I'm still trying to figure out whether it is possible to run RN directly in the browser using an interpreter instead of an emulator, along with something like emscripten (might have to run an emscripten-compiled JVM, too haha). Sounds really tricky, for one thing, any Android or iOS SDK calls would need intercepted and re-written for the browser, much like what WINE does.
There's this project: https://github.com/necolas/react-native-web that's similar to what you're describing. The downside is that you don't have access to any of the native APIs. Since we have an actually app we're able to support contacts, camera views, geolocation, etc.