is there a way to auto-increment the version code each time you build an Android application in Eclipse?
According to http://developer.android.com/guide/publishing/v
If you're using gradle then you can specific versionName and versionCode very easy in build.gradle. You can use git commit count as an increasing number to identify the build.
versionName
versionCode
build.gradle
You can also use this library: https://github.com/rockerhieu/Versionberg.