Mancini missed his side's second goal in the 2-0 quarter-final win, and was sent off, as he got involved in the stoppage time row with Sarri and was clearly livid when he spoke to RAI television a few minutes later.
"He should be ashamed, a 60-year-old man behaving in that way" said Mancini. "A man like him should not be involved in football."
Mancini also said that Sarri was "a racist" without giving further details on why he had made the claim.
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Sarri told RAI that he could not exactly remember what he had said but that the claims of homophobic insults seemed to be an exaggeration.
"This are things which happen on the pitch and should stay on the pitch," said Sarri. "I apologised to Mancini in the dressing room but he didn't accept it because he was still angry."
The row appeared to begin after the fourth official mistakenly indicated nine minutes of added time before changing his mind and indicating five instead.
Stevan Jovetic and Adem Ljajic scored in the last 20 minutes to send Inter into the semi-finals while Napoli had Dries Mertens sent off late in the game for diving.
Serie A leaders Napoli rested topscorer Gonzalo Higuain but Inter's Samir Handanovic was still the busier goalkeeper, making several last-ditch saves to keep Napoli at bay.
But Inter went ahead when Jovetic curled in a right-foot shot from 20 metres from a counter-attack in the 74th minute.
Mertens was given a second yellow card for a theatrical fall in the penalty area as Napoli pressed for an equaliser, before Ljajic broke clear from the halfway line to score Inter's second in stoppage time.
(Writing by Brian Homewood in Berne; editing by Justin Palmer)
