"All those responsible for this killing must be held to account including those who bear command responsibility," Jeremy Laurence, spokesman for the U.N. rights office, told reporters in Geneva.
Abakar was killed hours after he made remarks critical of the Sudan Rapid Support Forces paramilitaries in a telephone interview with a Saudi TV channel on Wednesday.
U.N. rights chief Volker Turk "is appalled by the killing", Laurence said.
He pointed out that the killing on June 14 happened "just hours after he was arrested by the RSF in El Geneina capital of West Darfur, where the conflict has taken on an ethnic dimension."
"Alongside liability of the direct perpetrator, Governor Abakar was in the RSF custody, and it was RSF's responsibility to keep him safe," Laurence said.
He pointed out that the governor was the second high-profile person killed in El-Geneina in a matter of days, after the older brother of the traditional chief of the Masalit, Tariq Abdelrahman Bahreldin, was killed.
Laurence also voiced deep concern at rising hate speech in the region, warning "it could further inflame tensions."
The U.N. rights office, he said, had reviewed video clips recorded by members of Arab militia boasting about "victories" and killing and expelling members of other groups.
"We call for justice and accountability for extra judicial killings, and all other violations and abuses that have occurred during this ongoing conflict," Laurence said.