The Three Lions beat the underdogs 2-1 after extra time in the Euro 2024 round of 16.
Slovakia were a minute from progressing, but were denied by Jude Bellingham and then a Harry Kane extra time winner.
Calzona said: “Rice was supposed to go to the referees and say goodbye and then leave.
“I had to speak to the referees and he was carrying on, continuing to speak. Then he apologized and it all ended there.”
"I have no bad words to say because we played a great game against an opponent that is a candidate to win it all,” the Italian added.
“My team showed a lot, we were really good, we conceded very little and we came within a step of qualification. Too bad.
"It's a pride to see a team like ours put in a performance like that. I can only say that for me it is a great pride to coach such a team.
“Fantastic group, a unique commitment and attitude. I was lucky enough to meet these guys and they gave me enormous satisfaction."