Conte became Napoli's fifth coach in little more than a year when he was hired in June and on Sunday he saw his new team crumble 3-0 at Hellas Verona in his first match back in Serie A.
"We melted like snow in the sun after the first goal," Conte said. "We should apologise to the Neapolitan people, who follow us with passion. I am the coach and it is right that I take full responsibility. In the second half there was an unacceptable performance.
"We should be ashamed, and I have to be ashamed as I am coach. Those who know me know that today my heart is bleeding, and I hope that some of the players are also bleeding so that would mean that at least we are on the right path."
Dailon Livramento and Daniel Mosquera - twice - scored on their league debuts to hand Verona the win on the opening weekend of the Italian league season and there was further bad news for Napoli as they lost key player Khvicha Kvaratskhelia to injury.
Napoli have been in a slump since Luciano Spalletti left last summer after steering the club to their first title in more than 30 years. The southern team finished 10th last season and hoped for much better under Conte, who led Juventus and Inter Milan to the title.
However, they drew 0-0 with Serie B side Modena in the Italian Cup on Conte's debut, going through on penalties, before Sunday's second-half horror show.
Conte's side had the better of the early stages in Verona and had chances with Matteo Politano, André-Frank Anguissa and Stanislav Lobotka before the home side almost gifted them the lead in first half stoppage-time.
Verona defender Martin Frese played a woefully short back pass to goalkeeper Lorenzo Montipò and Kvaratskhelia raced onto it but Jackson Tchatchoua came flying in for a heroic last-ditch tackle.
Kvaratskhelia had to come off immediately afterward, appearing to be indicating that he was feeling dizzy.
Verona came out a different team after the break and Grigoris Kastanos curled just past the far post with their first real chance before the hosts took the lead when Livramento managed to get enough on Darko Lazović's cross to poke the ball into the bottom right corner.
Verona doubled their lead when they won the ball in a midfield tussle and Ondrej Duda played it through for substitute Mosquera to fire past Alex Meret in the 75th.
Mosquera had only been on the field for less than two minutes. The Colombian forward doubled his tally in stoppages as he was left completely unmarked by the Napoli defence.
Verona were one of only two teams to win so far in the opening round. The other was Lazio, who beat newly-promoted Venezia 3-1.
In Sardinia, both Roma and Cagliari hit the woodwork and had other chances but could only draw 0-0. Bologna drew 1-1 at home to Udinese.
There were four draws on Saturday, including for defending champions Inter Milan and AC Milan.
Juventus host Como on Monday and Atalanta visit Lecce.
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