What would actually happen is that the US would start seriously threatening (blackmailing) the EU to a degree where it's forced to relent long before Apple would pull out.

Apple's estimated operating profit from the EU is around $40 billion dollars. If the US government wouldn't get involved, they could force Tim Apple himself to live on top of the Alps and he'd happily do it rather than lose that $40 billion, or shareholders would vote him out ASAP.