Despite Abete claiming he would try to convince Prandelli not to quit his post, the Italy handler was unequivocal.
"I will not go back on my decision," said Prandelli after seeing his side crash out of the competition following a 1-0 defeat to Group D rivals Uruguay.
Prandelli had taken over Italy following their first round exit from the World Cup in South Africa, steering them to a runner-up place at Euro 2012 and third place in last year's Confederations Cup.
But after failing to claim the draw that would have secured their ticket to the last 16, ten-man Italy had to settle for their second successive group phase humiliation.
Prandelli had extended his contract recently to take him up to the 2016 European Championships.
But the 56-year-old, who hit out at a lack of support from the country's own supporters, said: "Something has changed since I signed my contract."
He added: "We're the only national team that leaves (for the World Cup) without the support of its own people, who then rely on us to carry the hopes of a nation.
"When we left (for Brazil), we were jeered, whistled at and insulted. We were almost embarrassed to come here."
Abete said his decision had been made prior to the tournament, which saw Italy fail to reach the second round for the second successive edition.
"I'd already taken this decision prior to the World Cup," said Abete. "It's time for me to step aside. I think we need time to reflect."
Prandelli said he would accept all the "responsibility" for Italy's failure. But he called for sweeping reforms if Italy are to avoid falling behind the South American teams who have lit up the tournament so far.
Asked about the relatively poor performance of striker Mario Balotelli - who was substitiuted at half-time - Prandelli added: "Balotelli - that's a technical issue. I chose him, I chose a certain technical plan and it did not work.
"I take all responsibility for our failure, which, among other things, is down to our limitations in quality.
"It's useless for me to sit here and say I could have done it differently. This is why I'm resigning."
He added: "We knew it would have been difficult in conditions that would have allowed us to match certain teams.
"When Uruguay had possession, they were fast and we just didn't have that kind of speed. Maybe it worked with England, but it didn't work with the two other teams."
Italy began their campaign in positive fashion, but after a 2-1 win over England in Manaus a sluggish Azzurri were stunned 1-0 in Recife by Costa Rica.
Although requiring only a draw in Natal on Tuesday, the Azzurri were left with ten men after midfielder Claudio Marchisio saw red in the second half for a studs-up challenge.
The end of the game was also marred by claims that Uruguay striker Luiz Suarez sank his teeth into the shoulder of defender Giorgio Chiellini.
Prandelli said: "I did not see the incident. I did see the bite marks on Chiellini's shoulder."
He was more vociferous about Marchisio's sending off, which he claimed changed their whole tournament.
"You have to accept defeats, but when they are decided by wrong decisions it is hard to accept," he added.
"There was no need to show a red card. It caused a downward spiral that changed the match and led to us being eliminated from the World Cup."