The 'Springsteen' singer was due to perform at Coca Cola Field on Friday (June 1), but fans were forced to endure over an hour of delays due to severe rain.
Organisers eventually chose to cancel the event at 11pm when the rain showed no sign of yielding, angering some fans.
Some in the audience apparently then began tossing chairs onto the stage and smashing bottles throughout the arena.
Concert organizers later claimed that they chose to cancel the show over fears of a repeat of last year's stage collapse at Sugarland's Indiana State Fair gig.
Church issued a public apology to Buffalo News, saying: "I'm very bummed the show got cancelled... I've learned in doing this that some things are out of your hands and this is one of those things.
"I owe the city of Buffalo one, and I intend to make good on that. To everyone out there, thanks for coming and standing in the rain and wind, and we intend to bring it when the time comes."
Church is next scheduled to perform on Thursday, June 7 in Nashville.
Watch Eric Church's 'Springsteen' music video below: