Can I build Android apps with react native?
Yes.
React Native for Android was released September 14th, 2015.
No. You can't build Android apps with this release.
Facebook has indicated that they are working on an Android version, and famously said "give us 6 months", which some people have taken as a promise. I wouldn't base any important business decisions on the illusion that it will be unveiled when the 6 months is over, but you can at least take some hope in the fact that they're serious about making it production-ready before they give us access to it, and that it's not just a pie-in-the-sky hope.
Also, as others have pointed out, Facebook has already release apps that they've built with the Android version of React Native. So at least parts of it are ready for prime time.
UPDATE
Early support for React Native for Android has been pushed to GitHub!
As noted, there are several things that are currently not working if trying to port an application from iOS. But it's worth checking out.
https://github.com/facebook/react-native/commit/42eb5464fd8a65ed84b799de5d4dc225349449be
It is not aiming to be a cross platform, write-once run-anywhere, tool. It is aiming to be learn-once write-anywhere. -Colin Eberhardt, from Raywenderlich
Good news is that React Native for Android is finally here. (https://code.facebook.com/posts/1189117404435352)