Alekseyev was speaking after a meeting with Japanese Deputy Foreign Minister Yasuhisa Shiozaki in Moscow.
The fisherman was shot dead when Russian border guards detained the ship on August 16 -- accusing its crew of poaching, smuggling, and illegal entry into Russian territorial waters near the disputed southern Kurile islands.
Tokyo, which claims the islands as its Northern Territories, says the fishing boat was in Japanese waters at the time of the shooting. It says Russia's reaction was too severe and could harm bilateral relations.
Shiozaki said the fisherman's body will be handed over to Japanese authorities by August 19.
According to the prosecutor for Russia's Sakhalin region, the fishing boat's captain is the only crew member facing charges of illegal fishing and border violations. Such charges entail fines and jail sentences.
(AP, Reuters, Interfax)