Say one of the currently powerful countries establish a research station on the other side of the Atlantic, and it becomes self-sustaining at some point, from s

Say one of the currently powerful countries establish a research station on the other side of the Atlantic, and it becomes self-sustaining at some point, from say mining and wood harvesting and growing food on fields, powered by horsepower. What if that colony decides to become politically independent? Would any of the nations back on this side of Atlantic be able to project force on the far side of Atlantic? Won't it be very easy for the Americans to just shoot down any approaching ships?