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As an example, the Anchorage Air Route Traffic Control Center's sector extends all the way to the North Pole, far beyond 200 nm of the US coastline. Suppose that the US decides to "close its airspace" to Russian aircraft…
Would Russian aircraft still be allowed to fly in Anchorage's sector near the North Pole, which is in international waters?
Can ATC issue instructions to tell Russian aircraft to turn around? If so, by what reasoning?
Q: Can ATC issue instructions to exclude certain aircraft over international waters?
No. Live as of writing this, below shows a Russia-operated plane snaking its way through the Baltic Sea.
— Flightradar24
Civilian FIRs that extend beyond the territorial waters (12 nautical miles) are for traffic coordination, but when a country is banned from overflying the actual borders, international waters and EEZs are fair game, which are always inside somebody's FIR.
Exclusive Economic Zones (EEZs) only have material effect on maritime resources, including the sea bed, and do not have any legal consequences for aviation (UNCLOS Articles 57 and 58).
— International Airspace and Civil/Military Cooperation. ICAO. (PDF) [emphasis added]
6-Mar update: With Canada, USA, Denmark (inc. Greenland), and Iceland now not permitting Russian planes, below is an example of an Aeroflot flight to Dominican Republic where the Boeing 777 threaded its way between Greenland and Iceland (through one or both FIRs), and flew through the Western North Atlantic that is managed by Canada (Gander FIR) and USA (New York Oceanic FIR). (An earlier flight overflew Greenland, Canada, and USA.)
— Flightradar24
Another case is the 2017–2021 airspace closures to Qatari planes:
Above shows how Bahrain FIR – which is disproportionately big – extends to share borders with UAE and Iran FIRs; by staying away from territorial waters, Qatari planes could just fly through.
Notes:
The Qatari case went to the International Court of Justice, which judged ICAO should rule in the matter, but then the airspaces were reopened before it got that far (un.org; July 2020).
Military planes also do it all the time, e.g. like how Russian military planes have been going to the edge of UK's borders for many years now (here's one from 2015), cutting through many civilian FIRs while doing so.
Could be of interest: passing by Denmark into the Baltic Sea is open for all by the Copenhagen Convention of 1857.
Sure, US ATC can tell Russian aircraft that they're not allowed. Countries coordinate traffic outside their borders all the time. (Mostly over the ocean of course, but here's a video of a Canadian pilot, flying in Canada, who had to get permission from a US ATC facility to fly where he wanted.)
When in (or over) international waters, any vessel (or airplane) is considered to be under the laws of the country in which it is registered. Member nations of the ICAO have treaties with each other saying that airplanes will obey foreign ATC when flying in their jurisdiction, even if that jurisdiction extends beyond that country's borders. So, if a Russian airplane tries to fly through US-controlled airspace, the US says no, and they do it anyway, then, at least technically speaking, they're violating Russian law and can be prosecuted in Russian court for it (or however they handle aviation violations over there).
Of course, with the current tension between the US and Russia, Russian law enforcement are going to be a lot less likely to go after their own citizens because the US complained about them. The other side of that coin is that such planes are (slightly) more likely to be intercepted by US fighters, and the pilots of those fighters are going to be far more keyed up and ready to shoot if the violating plane doesn't comply with their instructions.
