Patrick Redmond, 68, from Coundon, Coventry, was assaulted at Rileys, in Hertford Place, just before 23:00 GMT on Monday.
He was taken to hospital with a serious head injury and his life support was switched off on Tuesday, West Midlands Police said.
A 37-year-old man from Radford, held on suspicion of wounding, is now being questioned on suspicion of murder.
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In a statement, Rileys said it was "extremely shocked by this tragic event" and it had provided police with CCTV footage and statements from staff.
It said its thoughts were with the family of the customer who died.
Rileys said: "We take the safety of our members very seriously and have a comprehensive zero tolerance policy towards violence.
"We will work with the police, and do everything we can to help ensure those responsible are brought to justice."
The police force said it was in the early stages of its inquiry.
Ch Insp Ed Foster said: "This is a tragic case and our sympathies go to the family of the victim."
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