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".
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