The call for an investigation was posted on the Foreign Office website, RIA Novosti reported.
"We are calling on the South Sudanese government to conduct a full-scale investigation into the accident, punish those guilty and implement all measures to guarantee such things do not happen again," the ministry said.
Four Russians died Friday when the Mil Mi-8 helicopter was shot down in Jonglei province. The United Nations said in a statement the helicopter was doing an aerial reconnaissance.
Philip Aguer, a spokesman for the Russian military, said South Sudan has apologized and says its soldiers believed the helicopter was carrying arms to rebels. U.N. Secretary-General Ban Ki-moon, who also called for an investigation, said the helicopter was clearly marked as a U.N. aircraft.
A U.N. mission in South Sudan was launched when the country became independent from Sudan last year.