Rafael Benitez's side fell behind in the 72nd minute of Sunday's Serie A meeting, when Mauricio Pinilla dispossessed Henrique and rounded goalkeeper Mariano Andujar to score.
Napoli's players protested that Henrique had been fouled, and although they did find a late equaliser through substitute Duvan Zapata, the club tweeted an angry reaction after the match in summing up Calvarese's display.
"It is shameful!" they posted. "[FIGC president Carlo] Tavecchio cannot lose credibility in this way.
"The refereeing was not fit for Serie A and falsified the league. In England this would not have happened."
Benitez was also scathing of the officials' performance after the match.
"I am convinced that the referee, the linesman and the fourth official saw the foul," he said. "It is difficult to officiate with only six referees, maybe they should add another two to see what happens."
Henrique, at the centre of the controversy, added: "I protested because I hadn't heard the referee's whistle, when I saw that a goal had been given I did not believe it."
Calvarese had a busy night, showing a total of 10 yellow cards, including two in the space of 21 minutes for Alejandro Gomez as the Atalanta striker was sent off after the break.
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