The trophy, which is given to the winners, was accidentally knocked over by a photographer during the media event in the Highland city.
A figurine of a footballer on top of the lid was knocked off and broken.
The original Scottish Cup trophy was made in 1873 and is the oldest surviving football trophy in the world; it is kept at Hampden while a replica is given to the winners each year.
The Scottish Football Association will repair the cup before the Hampden clash between Inverness and Falkirk on May 30.
Inverness tweeted after the event: “Due to circumstances beyond our control we have had to cancel tomorrow’s Scottish Cup visit to Falcon Square, Inverness.”
The club then added: “For the avoidance of any doubt, none of the #ICTFC players or club personnel were responsible for the damage done to the Scottish Cup.”