The British destroyer 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 Defender, and later threatened to bomb British naval vessels in the Black Sea if there were further "provocative" actions off Crimea, which Russia annexed from Ukraine in 2014. The peninsula is still recognised internationally as part of Ukraine.
The Defender'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.
Owen said Britain and its allies and partners such as Georgia, which hopes one day to join the Western NATO alliance, were committed to ensuring "Black Sea regional security, stability and prosperity and Georgia's territorial integrity and sovereignty".
Russia said it had dropped bombs in the path of the Defender on Wednesday as it was passing by Crimea, in what Moscow considered an incursion into Russian waters.
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.