According to the Ukrainian navy, the naval vessels Berdyansk, Nikopol and Yani Kapu were carrying out a planned transfer from the port of Odessa to the port of Mariupol on the Azov Sea.
One vessel was hit by the strikes, injuring a crew member, the Ukrainian navy said.
Moscow is able to control access between the Sea of Azov and the Black Sea after it built a bridge that straddles the Kerch Strait between Crimea and southern Russian Federation.
According to the Ukrainian navy, "the occupiers' dispatch service refuses to ensure the right to freedom of navigation, guaranteed by worldwide agreements". The strait is crossed by the recently completed Kerch Bridge, connecting Crimea to Russian Federation.
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Image: A Russian border guard vessel apparently tries to stop a Ukrainian navy tug boat.
The Kerch Strait links the Azov Sea with the Black Sea.
Russian news agencies reported that Russia's FSB security service had accused Ukraine of not informing it in advance of the journey, something Kiev denied, and had said the Ukrainian ships had been maneuvering dangerously with the aim of stirring up tensions.
Russian media cited authorities as saying that Kyiv had sought to "create a conflict situation".
The FSB statement did not mention the ramming of the Ukrainian tugboat.
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Russia's ships "carried out openly aggressive actions against Ukrainian naval ships", the statement said, adding that the Ukrainian ships were continuing on their way "despite Russia's counteraction".
Ukrainian Navy spokesman Oleh Chalyk told Ukraine's Kanal 5 TV that the tugboat "established contact with a coast-guard outpost" operated by the FSB Border Service and "communicated its intention to sail through the Kerch Strait".
The incident came amid a confrontation at sea between Ukrainian and Russian vessels.
A Rossiya 24 correspondent reporting from the area of the Kerch Strait bridge near Crimea said Russia's tactic had completely blocked the bridge, which connects the Black Sea and the Sea of Azov, and that Russian military helicopters were circling overhead.
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