Oscar Bartholomew, 39, died in hospital on the eastern Caribbean island on Tuesday, 24 hours after being taken into custody in the town of St. David's.
Family members have accused officers of beating Bartholomew after he mistook a plainclothes female police officer for a friend and hugged her, lifting her off the ground while in front of a police station.
Bartholomew, along with his wife and a cousin, had stopped at the St. David’s police station on Monday afternoon because his wife needed to use the bathroom, his aunt Josephine de Souza said in a telephone interview.
Police dispute that account, however, saying Bartholomew was arrested after assaulting an officer.
Dunbar Belfon, superintendent of the Royal Grenada Police Force, said the female officer was in uniform at the time she came into contact with Bartholomew.
Belfon said Bartholomew was taken into custody after the altercation and eventually transferred to hospital.
De Souza and one of Bartholomew’s cousins, Shem Pierre, accused police of leaving Bartholomew bleeding in his cell for at least three hours before calling an ambulance at the insistence of his wife.
According to doctors, a blood vessel in his brain had burst, causing his death early Tuesday, Pierre said in a separate telephone interview.
Bartholomew and his wife had arrived in Grenada on Friday to visit family for the holidays, Pierre said.
Attorney Derick Sylvester, who is representing the family, said relatives were giving their statements to police Wednesday.
"We’re putting things in place," Sylvester said, adding that the family is trying to hire a pathologist from the nearby island of Trinidad to conduct an independent autopsy.
He said the preliminary results of a government-ordered autopsy were expected late Wednesday.
Grenadian Prime Minister Tillman Thomas has called for a thorough investigation, and he and other high-ranking government officials visited the home of Bartholomew’s relatives Wednesday.
Condolences also came from Canada's Department of Foreign Affairs, where a spokesman said consular assistance was available if required.
Local police, meanwhile, pledged a thorough and objective investigation.
"The force has a very good track record of being very impartial in terms of conducting similar investigations," Belfon said.
"We have been very forthright in determining whatever investigation that needs to be conducted, even though it is done against members of the same institution."