Miccoli was close to tears at a news conference in the Sicilian capital, a day after he was questioned for five hours by public prosecutors.
"I ask forgiveness of the whole city of Palermo, I ask forgiveness of my family who brought me up with values and respect," Miccoli said, wiping at his face with one hand.
Allegations that Miccoli called Falcone "mud" came out after the player was accused of extortion.
Palermo's anti-mafia unit served Miccoli with an official notice of investigation after wiretap evidence reportedly showed him using the son of a mafia boss to recover money owed to him by the owners of a nightclub.
La Repubblica newspaper reported that the evidence also showed that Miccoli insulted Falcone, who was killed in a huge explosion along with his wife and a bodyguard as they drove on the outskirts of Palermo in May 1992.
"I'm a footballer, not a mafioso," Miccoli, 34, told the news conference. "I'm against all the beliefs of the mafia.
"In the last few years, I wanted not just to be the captain of Palermo, but to be available to everyone. I hung around with people who I thought would be real friends, but I was wrong.
"After everything that's happened it's been three days since I've slept, because things have come out that I absolutely don't think and I have demonstrated that with my actions," he added.
Miccoli, who will be released by relegated Palermo once his contract runs out soon, said the interrogation on Wednesday had made him realise he needed to be more responsible.
"After five hours of interrogation a new Fabrizio Miccoli came out," he told the news conference. "I answered every question that was asked of me. Now I need to be reborn, to stop all stupidity, I need to grow."
Falcone, who was from Palermo, was one of Italy's most famous anti-mafia judges and played a part in bringing about a huge trial in which some 360 criminals were convicted in the late 1980s and early 1990s.
