Dunne is facing up to two months on the sidelines after undergoing surgery on a troublesome groin injury.
He played in all three of the Republic's matches in Poland and Ukraine and in statement Villa said: "Richard Dunne will also be out of action for six to eight weeks, having undergone a procedure on a groin injury which he sustained and was not dealt with during the European Championships."
But the FAI claim Villa were kept fully up to speed with the 32-year-old's condition during the event and they agreed with his treatment.
"Richard Dunne received the highest level of medical care during his time with the Republic of Ireland team," an FAI statement read.
"All of the medical treatment he received was carried out in full and regular contact, and with the full consent and agreement of, the Aston Villa medical team and so we are very surprised by these suggestions."