Trapattoni said that Kelly issued him with an "I play or I go home" ultimatum ahead of Ireland's World Cup qualifier against the Faroe Islands last October.
But the Reading defender denies those comments and says Trapattoni is attempting "to defame my commitment and loyalty to Ireland".
Kelly, who has won 32 caps for Ireland, said: "It is extremely disappointing for me to have to make this statement. I am simply shocked and disgusted with the untrue and unwarranted comments made by Giovanni Trapattoni.
"These hurtful and untrue comments have caused distress and upset to myself and my family who have supported me throughout my career and know the level of commitment and passion I have always shown in representing my country.
"I have proudly represented Ireland from the age of 16, and I have never shown anything other than 100 per cent commitment to my country and my team.
"I cannot and will not tolerate attempts to defame my commitment and loyalty to Ireland.
"Representing Ireland at senior international level has given me some of my proudest moments in my professional career, and for me captaining my country will always be my greatest achievement.
"It saddens me greatly that I should have to defend my good character and reputation in a profession I love, however, I refuse to allow anyone to question my passion for representing Ireland.
"I will continue as always to proudly be available for my country."