Shane Bryant, 29, fell ill during an arrest outside the Co-op in Market Street, Ashby-de-la-Zouch, Leicestershire, on Thursday night.
Mr Bryant was taken to Nottingham's Queens Medical Centre, but died around 36 hours later.
A 24-year-old man was arrested and released on police bail on suspicion of grievous bodily harm.
Mr Bryant was stopped by members of the public while the other suspect left in a vehicle, which was waiting outside.
His death is being treated as unexplained, but a post-mortem is due to take place in due course. Police have launched an investigation.
Getaway foiled Shane Bryant died later in hospital
A witness said: "Two guys in an Audi tried to rob the shop". One guy armed with a golf club got everything money-wise from the shop, but members of the public stopped him and restrained him for 30 minutes until the police came.
"He left behind two small children and a absolutely heartbroken family".
'It's so hard to say goodbye to someone who has been a massive part of my life and to explain to my son that you have gone and its goodbye forever'. I think it may well have been the other robbers making their getaway.
A friend of Mr Bryant, Serina Larvin, has set up a fundraising page, saying: "A much loved friend of our community got taken away in unexpected circumstances".
Sassa CEO was forced to resign - Scopa chair
He has left at the time when we were managing the transition from Cash Paymaster Services to the South African Post Office. Magwaza was very vocal about the SASSA grant payment debacle earlier this year, describing it as embarrassing at the time.
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