Not sure how that would play out, but it would make an interesting law review article. And a fun oral argument.I'm pretty sure how it would play out. The courts would apply the political question doctrine and say that Article IV, Section 4 is a "textually demonstrable commitment" of the question to Congress and the Executive. It is for them and not the courts to say what constitutes an "invasion" and what protection is warranted. Even though it would not be a lawsuit against the federal government, attempting to get a court to compel it to act — it would only be a justification of the state's acting in its own defense — the courts would refuse to interpret and apply that provision of the Constitution.
UPDATE: Glenn fights back on the political question doctrine. He connects Article IV, Section 4 to Article I Sec. 10. ("No State shall... engage in War, unless actually invaded, or in such imminent Danger as will not admit of delay.")
Arizona is not — yet, anyway — engaging in war, but it’s clear from this language that it’s constitutionally empowered to do so when invaded, even if the federal government does nothing (and perhaps even in the face of federal objection). Arizona’s legislation is passed in response to armed people coming across the border and killing Arizonans, which sounds rather like an invasion. If that’s the case, then lesser responses to invasion are, arguably, permissible as well in the face of federal inaction. What the courts will do with this is, of course, uncertain (and likely not tied very closely to the actual text of the Constitution!) but it’s certainly not a frivolous argument.An immediate military response to a sudden invasion that "will not admit of delay" is clearly distinguishable from the long influx of migrants to which the state has responded with a stringent policy of requiring and checking papers and deporting people. Arizona has adopted its own immigration policy, because it doesn't like the policy the federal government is following. But the federal government has complete power over immigration. This "invasion" concept is offered as a work-around to that power.
I think that if the Arizona policy were challenged and Arizona argued it had suffered an invasion that the feds wouldn't deflect, that the courts would say: It's not for us to decide what constitutes an "invasion." Congress and the Executive have already made their decision about that, and the Constitution makes that the final answer. As they say in political question doctrine talk, there's "the impossibility of a court’s undertaking independent resolution without expressing lack of the respect due coordinate branches of government... an unusual need for unquestioning adherence to a political decision already made... [and] the potentiality of embarrassment from multifarious pronouncements by various departments on one question."
At that point, the court would be back at the original barrier to the state's law: the exclusive federal power over immigration.