The fatalities at Killarney on Sunday puts the safety of jumps racing under growing scrutiny.
Jumps racing was suspended in Victoria, Australia, last week, pending a review this week to decide its future.
Animal Aid said it was angered by the reported comments of a course official who described the Killarney deaths as "just the law of averages".
The organisation wants a prompt and thorough inquiry by the Irish Turf Club, as well as an independent review of horse deaths on all Irish racecourses.
Animal Aid says Sunday's deaths are thought to be unprecedented in modern times in Ireland, and matches the death-rate seen during a single day at the 2006 Cheltenham Festival - an occurrence that prompted a full-scale inquiry by British racing's regulatory body.
It said as well as the six dead, it understood four other horses were injured during the opening day of the three-day Killarney meeting.
The horses that died were Panther Creek, Robin Bu Bois, Imperial Hills, Wishwillow Lord and Sonorra. The identity and death of the sixth horse has yet to be confirmed.
Animal Aid director Andrew Tyler said: "To pass off the deaths of six horses in a single day's racing as a statistical blip amounts to cold-hearted complacency.
"The Irish racing industry claims it cherishes and cossets its thoroughbreds but the official response to Sunday's carnage proves that these horses are seen by some as mere disposable commodities."