Later, press center of the Ukrainian Navy has confirmed that after leaving the 12-mile zone, Russia's coast guard ships opened fire at Ukraine's navy group.
But Ukraine said it had informed the Russians of its plan to move its ships through the sea to Mariupol.
A bilateral treaty gives both countries the right to use the sea, which lies between them and is linked by the narrow Kerch Strait to the Black Sea.
Russian military ships were also deployed near the bridge.
The Ukrainian navy said in a statement posted on Facebook on Sunday evening that one of its small artillery boats, the Berdyansk, was damaged and one crew member was injured. It has claimed the waters off Crimea after annexing the peninsula in 2014. Transit under the bridge has been blocked by a tanker ship, and dozens of cargo ships awaiting passage are stuck.
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The Ukrainian Foreign Ministry said in a statement that Russia's actions were a violation of the U.N. Charter and worldwide law, and pledged to "promptly inform our partners about Russia's aggressive actions".
The Kerch Strait links the Azov Sea with the Black Sea.
Russia's border guard service accused Ukraine of not informing it in advance of the journey, something Kiev denied, and said the Ukrainian ships had been maneuvering dangerously and ignoring its instructions with the aim of stirring up tensions.
The Ukrainian navy made the announcement in a statement late Sunday. In particular, Don border guard cruiser rammed the Ukrainian armored artillery boat, which led to the damage of the main engine, planting and accommodation rail, life float is lost.
Ukraine's foreign ministry added that they would liaise with their allies concerning Moscow's "violent acts", saying they "pose a threat to the security of all states of the Black Sea region, and therefore require a clear reaction from the worldwide community".
The FSB statement did not mention the ramming of the Ukrainian tugboat.