Ancelotti was dismissed on Tuesday night after guiding Napoli to the Champions League round of 16.
Gattuso admits he has been in contact with Napoli president Aurelio de Laurentiis since Sunday.
“It was an easy decision," began the 41-year-old at a press conference.
“Napoli have been a leading club in Italy and Europe for many years. They are going through difficult times now, but most of the players, a good 90 percent, are functional to the type of football I want to play.
“I tell no lies. The first time I saw the President was on Sunday, I travelled to the FilmAuro offices [in Rome] and took a six-hour car journey each way, because otherwise you'd catch me on the train or plane.
“I wanted to call Carlo last night, but I knew he was with his family. We spoke this morning.
“It was not an easy couple of days for me, as I knew that I had to clarify my decision with Carlo and also speak to the staff.
“Yet again, Carlo proved himself a wonderful person today. He was like a father for me, I had over 300 games with him as coach.
“I have always called him for advice during moments of difficulty in this new stage of my career and he was always very kind. He confirmed that today, he told me where the team can improve and once again showed himself to be a great man."