The monologue was to be shown on State broadcaster Rai over the weekend.
The call was made by lawmakers for the Democratic Party (PD), the 5-Star Movement (M5S), the Green-Left Alliance (AVS) and Azione.
Several opposition MPs also called on Meloni to express a “clear” position on anti-Fascism.
In the text of the monologue, Scurati described the “ruling party” as “post-Fascist” and said it was trying to re-write history rather than “repudiate its neo-fascist past”.
Scurati was set to present it in view of Thursday’s Liberation Day national holiday, when the country celebrates its liberation from Fascism and the Nazi occupation in World War II.
Meloni criticised Scurati in a Facebook post, saying she would never ask for censorship as she had been “ostracised and censored” by the public broadcaster in the past, and suggested Rai had simply refused to pay 1800 euros for a one-minute monologue.
She also published the text of the monologue “so Italian people can freely judge its content”.
On Sunday Scurati said he had been subjected to a form of “violence” by the prime minister and felt afraid.
“When the boss points the finger at the enemy and the newspapers put you on the front pages, they put a target on your face,” he said.
“Then there might be someone who takes aim at that target.”
Opposition parties criticised the decision to stop Scurati giving the monologue, saying it was part of an effort to limit dissenting voices and turn the State broadcaster into a “megaphone” for the government.
ANSA