Mr Fungavaka was in Tonga for his grandfather's funeral, and on Friday he went into central Nuku'alofa with a relative and was drinking at a club.
A Tongan police statement said he was arrested for minor drunkenness and taken to the central station, where he was assaulted by a prisoner in the cells the following morning.
As a result, Mr Fungavaka underwent surgery for a head injury and was put on life support. He did not regain consciousness.
Tongan media reported that Kalisitiane Manu, who was arrested on Friday for drunkenness in a public place, had been charged with "bodily harm of the victim" and would appear in the Magistrate's Court on Monday.
But two Tongan police staff have also been arrested for an earlier assault on Mr Fungavaka.
Assistant Police Commissioner 'Unga Fa'aoa would not be drawn on the police investigations, confirm the prisoner's name or identify the two officers, saying the matter was before the court.
NZ Deputy Commissioner Mike Bush paid tribute to an outstanding policeman whose bravery was recognised when, while still a civilian, he tried to save a woman who had been set on fire by her boyfriend.
"Kali's colleagues describe him as quiet and unassuming, very humble and very passionate about policing and making a difference, especially in the Polynesian community.
"He was always one of those staff members who could be relied upon when the going got tough. He was a valued member of the police family.
"We have conveyed our deepest sympathy to his partner and family and our thoughts are with him."
Mr Fungavaka did not buy travel insurance, Ms Watts said, but police friends had opened an account to help with costs to bring his body home.
Police Association president Greg O'Connor said his organisation would help financially; the extent of support was still being worked through.
He had spoken to Mr Fungavaka's colleagues, who were shocked at his death, at a time when they were reeling from the death of Otara mother Liku Onesi on Wednesday after a police vehicle on its way to a burglary collided with a four-wheel-drive in which she was travelling.
In 2003, before he became a police officer, Mr Fungavaka witnessed an attack in Otahuhu in which Ahmad Riyaz Khan set fire to his former girlfriend Gulshad Hussein, a 23-year-old Shell petrol station manager. Khan poured lighter fluid on her and lit it.
Mr Fungavaka raced across the road when he saw Ms Hussein trying to turn on a water tap to extinguish her blazing clothes.
He ran into the service station, which was on fire, to find an extinguisher, despite the risk of an explosion. He doused the flames on Ms Hussein and then ran back into the station to put out the rest of the fire. Tragically, she died in hospital from her injuries
In 2006, when the Government awarded him the New Zealand Bravery Medal, he told the Herald it was "pure instinct" to help, and the incident had inspired him to join the police.
SUPPORT FUND
Kali Fungavaka's colleagues have set up the Kali and Audra Support Fund at the Otahuhu National Bank. Account details: 06-0217-0352269-00