Soak grape leaves in a bowl of warm water, changing water every 10 minutes, while rice is cooking.
Bring water, rice, and cardamom to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until the rice is tender and liquid has been absorbed, about 30 minutes. Cool rice to room temperature, about 30 minutes. Discard cardamom pods.
Mix onion, persimmons, mint, salt, and pepper into cooled rice.
Lay 1 grape leaf onto a work surface with stem facing up. Spoon 2 to 3 tablespoons rice mixture into the center of each leaf. Fold the bottom part of leaf over filling. Fold sides over bottom fold and tightly roll leaf around filling toward the top of the leaf. Repeat with remaining leaves and filling.
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add stuffed grape leaves, cover, and steam until filling is fully cooked, 20 to 30 minutes.