The Status of Forces Agreement (SOFA) between the United States and another Country determines who handles what in these cases. For the US Navy, the Naval Criminal Investigative Service (NCIS) would investigate the incident. Every Aircraft Carrier has an Agent or Agents assigned from NCIS. NCIS maintains overseas offices as well. I did five years with NCIS.
Local Police would also conduct their standard investigation and could file charges in their Court System. The US Navy tries to shift the matter into their own system whenever possible.