"It's very sad. It could have been worse, Ryan could have taken a fall off him, someone could have been seriously injured".
French-trained horse Verema had to be put down after an injury suffered in the 2013 Cup.
In 2015, fan favourite Red Cadeaux was euthanised weeks after breaking his leg during the race.
Stier said the injury to The Cliffsofmoher was an unfortunate incident that happened infrequently, as Victoria had one of the best safety records in world racing.
Limb injuries are the most common cause of death for racehorses on the track.
There have been several deaths in the past five years of the glamour race.
UPDATES: THE horse that exited midway through the Melbourne Cup - the Aidan O'Brien-trained The Cliffsofmoher - has been euthanased.
In a statement, RSPCA Australia spokesperson Jane Speechley said: "Our thoughts are with animal lovers across the country who have been affected by this loss, and confronted by the very real risks posed by racing to the horses involved".
Veterinarians rushed on to the track as soon as the race was completed as a tarp was erected around the fallen stayer.
Cross Counter was the final victor of the Melbourne Cup after a nail-biting final, taking the lead in the very last seconds of the home stretch.
The UK thoroughbred appeared agitated moments before the race began, pulling up lame just as the field passed the finishing post in the opening lap.
The Chairmen and Directors of the Victoria Racing Club (VRC) later extended their sympathies, with Flemington Racecourse tweeting their condolences, "To the Williams family and the connections of The CliffsofMoher, which was euthanised after he went amiss in the Melbourne Cup".
"Cliffsofmoher, he's melting like an ice-cream at the moment", Seven commentator Richard Freedman said.
The $6.2 million race was won by Cross Counter, followed by Marmelo and A Prince of Arran.