Dave, for some light reading start with the entire USC and CFR, there will be a quiz on Tuesday morning. (You get an extra 24 hours because of the Columbus Day holiday.)
" I find it interesting that a US citizen would be expected to comply w/ a Navy order anywhere in the world, regardless of circumstances; what recourse would one have?" Yeah, well, there's the Constitution, the Posse Comitatus Act (partly repealed/amended last year in the Defense Omnibus Spending Act appendixes), and a host of other law that you should know about as a US citizen. It is your job to BE INFORMED if you expect to run our government.
You don't have to comply with military orders ANYPLACE in the world, only outside of the US. Within the US, military forces were barred from law enforcement duties by the PCA shortly after the Civil War (which changed many things in the US, slavery being the least of them) and now, they are still barred except by Presidential declaration of emergency. (A change snuck in by cowards who didn't understand that the military could indeed help after Katrina, and by military leaders who probably were right to say they wouldn't perform any disaster duties without their arms...a valid conflict.)
But none of this should be news to you. Of course, if you are a US citizen between the ages of 18 and 45? Surprise, you are also legally bound to federal Militia service. That's the basis of our draft, so that's not going to change. You get on your satphone and call your Congresscritter, and see if the Admiral can't get on his phone faster and have you placed on active duty and then in his brig.
Yes, there are all sorts of arcana in the laws and most still serve a purpose.
You expect a billion dollar fleet exercise to be scheduled after they do what? Send out engraved invitations asking yachties if it would be convenient?
You have no legal rights, unless you know and understand the laws. Kinda like all the folks who never should have been given mortgages they never bothered to read--it just ain't gonna end well that way.
Oh, and speaking of recourse? Search on "sovereign immunity". By and large, you can't sue the king or crown. Your options are highly limited, even in the Federal Republic of the United States of America.