The British destroyer HMS Defender, which angered Russia by sailing through waters off Crimea, docked at the Georgian Black Sea port of Batumi on Saturday.
Russia said it had fired warning shots at the destroyer, and later threatened to bomb British naval vessels in the Black Sea if there were further “provocative” actions in the area.
The dispute between Moscow and London erupted on Wednesday after the HMS Defender passed near Crimea, which Russia annexed from Ukraine in 2014. The peninsula is still recognised internationally as being part of Ukraine.
Russia's Foreign Ministry summoned the UK ambassador on Thursday to "strongly protest" against what it said was a British destroyer's breach of its territory.
Russia said it fired warning shots at the destroyer to ward it off, but Britain denied the claim.
The vessel’s commander, Vince Owen, said the British Navy was committed to “providing reassurances and security in the region, and incredible deterrence to those who seek to undermine global security”.
The ship has visited Istanbul in Turkey and Odessa in Ukraine as part of a tour of duty that will take its carrier group through the Suez Canal and on to East Asia.
Commander Owen said Britain and its allies and partners such as Georgia were committed to ensuring “Black Sea regional security, stability and prosperity and Georgia’s territorial integrity and sovereignty”.
Britain said the Defender had sailed through waters belonging to Ukraine, and that any shots fired had been part of a pre-announced Russian gunnery exercise. It said no bombs had been dropped.