The BBC said the stage problem was discovered before a sound check on Friday night, prompting concert officials to cancel that night's performance mere hours before it was scheduled to begin.
The next night's performance was quickly canceled as well and both have been rescheduled for next weekend, with original tickets still being valid for the new dates, the BBC said.
"Our most sincere apologies to all those who were supposed to attend these shows this weekend," Gilmour said of the situation.
"A structural defect was discovered in the scaffolding just before the sound check this evening, which has rendered the whole stage structure unsafe."
An incident in Venice's St. Mark's Square is nothing new to Gilmour, who saw nearly 200,000 fans overcrowd the square in 1989 for a Pink Floyd concert, causing excessive damage to the historic area, said the BBC.