Devin Jacob Ballard, 20, is suspected of killing Anna Yagi, 39, in a hotel in Naha, the capital of the southernmost prefecture, on Saturday and taking her wallet and backpack, police said.
Ballard has denied the allegations.
"The occurrence of such an extremely brutal and vicious incident is extremely regrettable," Takaichi said in a statement.
"The government of Japan strongly protests to the US side and strongly seeks maximum cooperation with the investigation," as well as the tightening of discipline and thorough measures to prevent a recurrence, she said.
The US embassy and the Marine Corps expressed condolences over the incident and said they were cooperating with the Japanese authorities.
The Marines said in a statement that the suspect, who is assigned to the III Marine Expeditionary Force and based at Marine Corps Air Station Futenma, was in Japanese custody.
"The Marine Corps takes these allegations very seriously and expects every Marine to adhere to the highest standards of behaviour," it said.
US Ambassador to Japan George Glass wrote on social media that he had called Okinawa Governor Genta Koja, offering sincere condolences and pledging full cooperation as the investigation continued.
Most of the 18,000 Japan-based Marines are stationed in Okinawa.
Their presence has repeatedly stirred resentment among residents over issues such as noise, aircraft crashes and the conduct of personnel.
Saturday's incident could complicate Takaichi's efforts to relocate some Marines from the busy Futenma air base to a more remote facility at Henoko.
Opponents want the troops moved out of the prefecture.
Okinawa, which is 100 km from Taiwan at the closest point, is viewed by Tokyo and Washington as key in deterring potential military aggression from China.
with Reuters
