Rice played the full 120 minutes as England came from behind to secure a 2-1 victory over Slovakia in the last 16, with Jude Bellingham's dramatic 95th-minute equaliser followed by Harry Kane's header just 52 seconds into extra time.
After the game, Rice was shoved in the chest by Calzona and both men were then involved in a fiery exchange in front of the match officials.
England's No.4 appeared to try and intervene when Calzona approached Turkish referee Halil Umut Meler after the full-time whistle, before reacting angrily to the Slovakian's push in the chest, sparking an X-rated exchange of words.
According to lipreader Jeremy Freeman, via the Daily Mail, Rice told Calzona to 'Shut up, p***y, oi, shut up, you bald c***.'
Calzona comments on heated Rice altercation
Calzona has since revealed that Rice has apologised, and the Italian stated that he initially wished to confront the officials at full time to complain about England's time-wasting tactics.
"Rice was supposed to go to the referees and leave," Calzona told reporters. "I had to speak to the refs and he wasn't leaving, he carried on. But then he apologised and it was all fine. I didn't like the way the England team were wasting time and not punished."
Calzona later insisted that he was proud of the performance of his Slovakian side, adding: "I have no bad words to say because we played a great game against an opponent that is a candidate to win it all.
"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."
Rice coy on Calzona incident, praises England's "spirit"
Meanwhile, Rice decided not to address the incident with Calzona and instead opted to praise the togetherness and never-say-die attitude of his England teammates in the extra-time victory.
"There's that inner fight and spirit in us [that] a few Slovakian players before the game said that we didn't have. We've got that togetherness, we proved that tonight," the Arsenal midfielder told reporters.
"We'd do anything to protect this manager (Southgate), protect each other. We'll keep going and keep fighting and what we've got as a group tonight showed out there. It's an honour to be a part of it and we're going to give it everything."
While Slovakia have been eliminated from Euro 2024, England will now turn their attention to a quarter-final clash with Switzerland at Merkur Spiel-Arena in Dusseldorf on Saturday.