It said in a statement that the Turkish fishing vessel, which it did not name, had failed to respond to earlier warnings, but had sharply changed course after shots were fired before passing within just over 500 meters of the warship.
The incident, which occurred on Sunday morning, is likely to heighten tensions between the two nations who are seriously at odds over Syria and the Turkish shooting down of a Russian military jet last month.
Russian President Vladimir Putin, who called the downing of the plane a "stab in the back," has since imposed economic sanctions on Turkey as a retaliatory measure.
Earlier this month, Turkey complained to Russia over an incident in which a Russian sailor was pictured brandishing a rocket launcher on the deck of a naval ship passing through Istanbul.