The ex-chief executives, who previously ran Tesco, Sainsbury's, Asda, Morrisons, M&S and B&Q, said a drop in the pound coupled with supply chain disruption would cause prices to spike.
The average household could be £580 a year worse off as a result, retail union USDAW has predicted.
Vote Leave rejected the claims.
"Independent experts are clear, EU membership and the meddling court that comes with it have put up prices and hurt our economic growth," said Vote Leave chief executive Matthew Elliott.
"The EU is costly, bureaucratic and blind to the impact it has had on people's wages and soaring energy bills."
But the retail bosses, which include former Marks and Spencer boss Marc Bolland and former Sainsbury's chief executive Justin King, insisted that a UK exit from the European Union would be "catastrophic for millions of ordinary families".
USDAW said it had calculated its £580 figure based on an expected fall in sterling in the event of the UK's exit from the EU, together with expected new tariffs imposed on imported EU goods including food, drink and clothing.
The former bosses of Tesco, M&S, Sainsbury's and B&Q already warned last month that prices would rise amid a so-called Brexit, but their ranks have swelled with the addition of the ex-chief executives of Asda and Morrisons.
Richard Lloyd, former executive director of consumer group Which?, said he was now "convinced that leaving the EU will give ordinary British families a worse deal for years to come".
"My advice to consumers is clear - this could be an expensive mistake, don't risk it," he said.
Tough competition from discount rivals together with low inflation means that overall the prices of groceries has steadily fallen.
According to consumer research firm Kantar Worldpanel the price of groceries has fallen every month since September 2014, with the average household now spending £78.10 a week in the supermarket.

The 26-year-old ex-Dagenham and York defender started all but four of Wood's league matches last season.
Meanwhile, striker Ryan Bird has joined from Newport, signing a one-year deal with an option of a further year.
The 29-year-old scored six goals in 19 appearances for the League Two side last season.
The duo are Chris Kinnear's seventh and eighth summer signings, with Tobi Sho-Silva and Connor Essam among those to arrive at the Crabble Athletic Ground.

The 86-year-old first became prime minister in 1982, beginning a third spell as leader in 2014.
In a televised address Mr Jugnauth said he was resigning in favour of a "younger and more dynamic leader". His son is currently finance minister.
Opposition parties have criticised the move but in practice there is little they can do to stop it.
The younger Jugnauth is head of the Militant Socialist Movement, the largest party in the governing coalition.
A former prime minister, Navin Ramgoolam, warned that the family was turning Mauritius into a "banana republic".
Mauritius, an Indian Ocean archipelago, is a former British colony.
In his speech, Mr Jugnauth cited the "Westminster tradition", referring to the transfer of power from David Cameron to Theresa May in the UK without elections, local media reported.
He is due to hand in his resignation on Monday morning.
The price of everyday essentials such as food, drink and clothing would rise if the UK votes to leave the EU, former retail bosses have warned.
[NEXT_CONCEPT]
Dover Athletic have signed Femi Ilesanmi on a one-year contract after the left-back's departure from National League rivals Boreham Wood.
[NEXT_CONCEPT]
Mauritian Prime Minister Anerood Jugnauth says he is stepping down to hand power to his son, Pravind.