The vandals also broke down the gate to the bridge leading to the temple.
Russia has faced a much larger resistance than expected in Ukraine and is being pushed back in some regions.
Thee local authorities said they were not aware of the situation.
Moscow isn't tolerating any dissent in this matter.
On July 12, the metropolitan organised an ecumenical memorial for the fallen Ukrainian soldiers at Lviv's Lychakivsky Cemetery.
The property damage caused by the fire is estimated to be more than USD 170,000.
The monastery lies near Russian positions in eastern Donetsk, which the Kremlin is focused on capturing.
The situation in Crimea is an example of the hostile treatment by Russian authorities against religious minorities.