First: What is the “Militia”?
Webster’s American Dictionary of the English Language (1828) tells us:
“The body of soldiers in a state enrolled for discipline, but not engaged in actual service except in emergencies; as distinguished from regular troops, whose sole occupation is war or military service.
The militia of a country are the able-bodied men organized into companies, regiments and brigades, with officers of all grades, and required by law to attend military exercises on certain days only, but at other times left to pursue their usual occupations.”
Article I, Sec. 8, clause 15 grants to Congress the power to provide for calling forth the Militia to [among other things] “repel Invasions”.
So! One of the functions of the Militia – that body of weekend warriors trained by the States and whose officers are chosen by the States, is to defend the States against Invasions.
But what if the federal government refuses to act?
Alexander Hamilton provides the answer in Federalist No. 29.
“…it would be natural and proper that the militia of a neighboring State should be marched into another, to resist a common enemy…
True, it was contemplated that the “United States” would normally be the entity which protects the States against Invasion (Art. IV, §4). But when the federal government has demonstrated its determination that the States are to be overrun by invaders, then the States are within their Retained Sovereign Rights to employ the Militia to defend their People from those into whose hands the federal government has demonstrated its determination to deliver them.”
So, clearly, the Sovereign States may use their State Militias and engage in War to defend themselves from the Invasions.
Notice Hamilton did not suggest lawsuits as the answer!
Neither did Madison. Neither did Jefferson. Etc.
For more study (and there is much more), go here: https://publiushuldah.wordpress.com/2010/05/04/the-invasion-of-arizona-the-remedy-when-the-federal-government-refuses-to-do-its-duty/