NATO ships in theory can open fire on the Russian fleet in the Black Sea. This was stated by the publishing Business Insider Commander of the US Navy in Europe and Africa Admiral Robert Berk.
“If Russia has a provocation against the alliance forces, and the Captain of the NATO ship will account for this as a threat, it will strike. I am not going to demand the first blow to the jaw from my commanders,” the American admiral said.
General Robert Berk noted that NATO will not shoot first: “The ships and aircraft of Russia constantly accompany NATO forces in the Black Sea, and usually the actions of the Russian military are professional and look acts of aggression.”