A Ukrainian navy statement says the "openly aggressive actions" occurred as three of its vessels sailed to Mariupol in the Sea of Azov.
Its tug, named Yana Kapu, allegedly suffered damage to its engine, guard rail and outer shell.
But Russia said the ships were in its waters, and accused the Ukrainians of "provocative actions".
The Sea of Azov lies east of the Crimean peninsula, which was annexed by Russia in 2014, and south of regions held by pro-Russian separatists.
Russia said the Ukrainian ships - two warships named as the Berdyansk and the Nikopol, and the tug - were trying "to create a conflict situation in this region".
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