It's more subtle than that. The way United States copyright law is structured is that it tends to have simple, broad rules for what is subject to copyright, and then has several exceptions that limit the copyright owner's exclusive rights.
The tendency is to put the complexity and special cases in those exceptions and to keep the basic rules for what is subject to copyright simple.
Under current law and current Supreme Court precedent, for written works (including computer programs) that standard is essentially that if it is original and it has some creativity it is subject to copyright. The level of required creativity is very low. I believe the Supreme Court used the phrase "modicum of creativity" or something similar.
Microsoft's argument was that APIs can qualify under these simple, broad rules as interpreted by current precedent, and so the district court went too far in saying that APIs cannot be subject to copyright.
They did not argue that this means that Oracle should be able to prevent Google's copying. They believed that one or more of the exceptions (fair use, the merger doctrine, and some others they mention in their brief) could apply.
Their argument on destabilizing the software industry didn't have anything to do with APIs specifically. Essentially it was that the reasons the district court used to exclude the APIs from being copyrightable would likely apply to much more, calling in question a large part of the idea of copyright for computer programs. That would undermine pretty much all software licensing, both proprietary and open source.
It's worth reading their brief, as I've done some summarizing and simplifying that probably distorts some of the arguments. The Ars article you cited links to it.