Adrian Ismay, 52, was seriously hurt after a booby-trap device exploded under his van in Belfast on 4 March.
The police said that a post-mortem examination showed he died as a "direct result of the injuries" of the bomb.
A dissident republican group, widely referred to as the new IRA, said it carried out the attack.
The organisation said Mr Ismay was targeted for training officers at Maghaberry Prison near Lisburn, County Antrim.
Sue McAllister, director general of the Northern Ireland Prison Service, spent time with the Ismay family on Wednesday.
She said they were "devastated and beyond shock at what's happened".
"Like all of us, they thought that he was on the mend and was doing really well," said Mrs McAllister.
"We thought he would make a full recovery, we expected him to come back to work in due course, so we're all shocked, but particularly Sharon and her daughters."
Mrs McAllister condemned the behaviour of some paramilitary inmates at Maghaberry Prison, who reportedly cheered, smoked cigars and mocked wardens at the high-security jail when news of Mr Ismay's death filtered through.
"This was just unacceptable behaviour by a very small number of prisoners," said Mrs McAllister.
"My colleagues in Maghaberry dealt with it professionally, with great fortitude and with the courage they display on a daily basis."
She admitted she and her colleagues in the Prison Service felt "very angry" on hearing the reports of the celebrations in the exercise yard of the prison.
"We encounter challenging behaviour by a small number of prisoners every day, and we deal with it with great professionalism and we do not stoop to behaving that way in response," she added.
A man appeared in court on Saturday charged in connection with the bomb attack on Mr Ismay.
Det Ch Insp Richard Campbell said it was "yet another difficult day for the Ismay family, his friends and colleagues as they struggle to come to terms with the events of the past 12 days".
He added: "We have spoken to the family this morning and advised them of this development.
"We are also liaising with the Public Prosecution Service in relation to the individual who is currently charged with attempted murder and causing an explosion with intent to endanger life."
The PSNI have also repeated an appeal for information on two cars that police believe may have been used by the bombers - a red Citroen C3 and a silver Skoda Fabia.

Robin Li was in the passenger seat when he made a video call to an artificial-intelligence developers conference in the Chinese capital.
But while autonomous vehicles are tipped as the future of motoring, they are banned from China's public roads.
Police began an investigation after the video went viral.
The footage shows another Baidu executive in the driver's seat, but with his hands off the steering wheel.
Mr Li reportedly told his audience the vehicle was on autopilot and that the journey was smooth.
Baidu, which has not commented on the incident, is one of many firms developing prototypes of self-driving vehicles that will use mapping and artificial intelligence technology.
At the conference, Baidu announced that it had built an alliance with 50 partners to develop and promote self-driving cars.
They include Ford, Daimler, five Chinese car manufacturers, mapping firm TomTom, and chipmakers Intel and Nvidia.
Baidu said it hoped to have autonomous cars on China's roads by 2019 - so long as the law allowed it - before expanding to other markets including the US.
Earlier this year Baidu also opened up its self-driving car codes to software developers via the 'Apollo' open-source platform in the hope it would spur innovation.
Two of its autonomous vehicles using Apollo version 1.0 were on display at the developers conference, the Wall Street Journal said.
Google parent company Alphabet, Ford and GM have also built and are among those testing self-driving cars.

The 18-year-old can play at right-back, centre-half and in central midfield.
Born in Birmingham, Suliman signed a professional deal with Villa in January 2015 until 2017, but is yet to make a senior, competitive first-team appearance for the Championship club.
He has represented England since under-16 level and made 16 international appearances for the under-17 side.
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The PSNI has launched a murder inquiry into the death of a prison officer who died 11 days after being injured in a dissident republican bomb attack.
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The head of Chinese online search giant Baidu is being investigated, after riding in one of the firm's driverless cars on a Beijing ring-road.
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Cheltenham Town have signed Aston Villa and England Under-18 defender Easah Suliman on loan until 2 January.