Whether or not we limit he discussion to this case, I don't think there's any reason to fear that using an API would be a copyright issue (though again IANAL). The whole purpose of an API is for people to write software against it. I think that people misconstrue what this case is really about and start talking in hyperbolic terms (not that you're doing so, but in the popular press, blogs, etc.).
At the risk of mischaracterizing Oracle's argument, the basic idea is that Google wanted developers to have a familiar API and wanted to reuse some of the tooling built for Java, but there is no way to get a compiled Dalvik program to run on a JVM, and likewise compiled Java software can't be run on Android. So Google piggybacked on the hard work Sun put into designing a coherent (and quite large) API, but didn't really do so in an interoperable way. In some ways the case is reminiscent of the Sun/Microsoft quarrel about Microsoft's extensions to Java (though that also involved trademark issues, as I recall), though many commenters who supported Sun in that case now seem to support Google in this one.
At the risk of mischaracterizing Oracle's argument, the basic idea is that Google wanted developers to have a familiar API and wanted to reuse some of the tooling built for Java, but there is no way to get a compiled Dalvik program to run on a JVM, and likewise compiled Java software can't be run on Android. So Google piggybacked on the hard work Sun put into designing a coherent (and quite large) API, but didn't really do so in an interoperable way. In some ways the case is reminiscent of the Sun/Microsoft quarrel about Microsoft's extensions to Java (though that also involved trademark issues, as I recall), though many commenters who supported Sun in that case now seem to support Google in this one.