There is nothing automatically secret in an instance of an Android app that your server could verify. Using Google/Facebook OAuth login is a reasonable choice.
You can use the Facebook SDK for Android to allow the user to log in using their Facebook account, and as a.ch. mentions in the comment you can do the same with the Play Services SDK to log in via their Google account.
In both of these cases, this works because there is a local broker app (Facebook, or Google Play Services) that is trusted by the identity server (facebook.com, or google.com). The broker app has the user's credentials, can verify the identity of the app that is requesting access, and can display a view where the user can approve/reject the request. Once the identity server knows that the user has logged in on the device, knows the identity of your app, and knows that the user approved the login request, it can safely send back a token to your app via the broker. That token allows whatever permissions your app requested and the user approved.
Your WebView can then include that token in an initial request to your rails server, which checks the token with Facebook/Google. From this point, the code should look similar to the existing desktop web, at least with respect to user authentication.
You can download the Facebook SDK for Android from https://developers.facebook.com/android/, and you can get the Google Play services SDK in the Android SDK manager ('android sdk') under Extras. Both have tutorials to help you get started.