The Azzurri fell at the play-offs again, this time after a penalty shoot-out against Bosnia and Herzegovina on Tuesday, and will miss this summer's finals in the United States, Canada and Mexico.
The head of the federation (FIGC), Gabriele Gravina, said he would not resign but a board meeting next week would decide whether he will keep his position.
"It's clear that Italian football needs to be rebuilt from the ground up and that starts with changes at the top of the FIGC," Abodi said in a statement.
Gravina has been in charge of the FIGC since late 2018, when he was elected to be the permanent replacement for Carlo Tavecchio who stepped down after Italy's first World Cup play-off disaster against Sweden the year before.
A FIGC source confirmed to AFP that Gravina would hold a meeting on Thursday afternoon with the heads of Italy's three professional divisions, and both the the players and referees' associations.
Some fans made their feelings known about Gravina and the FIGC by chucking eggs at the governing body's headquarters in Rome.
Abodi's comments come amid hostilities between the Italian government and Gravina, who when speaking to the media after Italy's defeat lashed out at a perceived lack of support or football from the state.
Gravina admitted that Italian football "is in a profound crisis".
"It is a crisis that requires serious reflection and not only by the FIGC but also from politicians who have only pushed for resignations," said Gravina.
He also referred to other sports as "amateur" and "state sports" compared to football due to the large number of athletes, particularly Olympians, who are nominally employed by different arms of Italy's armed forces and police.
Italy claimed a record 30 medals at the recent Milan-Cortina Winter Olympics, including 10 golds, and left the 2024 Summer Games in Paris with 40 medals.
The Mediterranean nation also has top performers in a wide variety of other sports, with tennis star Jannik Sinner, a four-time Grad Slam winner, the most obvious example.
"I believe it is a mistake to deny responsibility for the third missed World Cup qualification and accuse the institutions of a presumed failure while downplaying the importance and professionalism of other sports," added Abodi, who has been sport minister in Giorgia Meloni's hard-right government since 2022.
Speed skater Francesca Lollobrigida, who won two Olympic golds this past winter, was one of several athletes to react to Gravina's comments, sarcastically saying on Instagram "I'm an amateur".
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