"Thanks for everything, coach," the club wrote on their social media page.
Napoli ended their season with a 1-0 home victory over Udinese on Sunday, which secured a second-place finish behind Serie A champions Inter Milan. Conte announced he was stepping down at a press conference before clashing with the president.
De Laurentiis claimed that Napoli would have defended their league title had his squad not been depleted by fitness issues throughout the campaign.
Napoli struggled for a large part of the latter half of the season, with key players such as Scott McTominay, Kevin De Bruyne, Romelu Lukaku and captain Giovanni Di Lorenzo sidelined for long spells.
"Without the injuries, we would have been the champions," De Laurentiis said.
The remarks drew an immediate response from Conte, sitting alongside the president, who urged greater respect for Inter's achievements.
"That probably would not have happened," Conte said. "Inter's merits need to be recognised.
"I'm sorry, but I don't like clinging to these things. We have to acknowledge others' victories and show respect. If we want respect, we have to give it. Inter deserved to win the league, with injuries or not."
The duo revealed that the decision to end their professional relationship had been reached months earlier in the campaign.
De Laurentiis refused to name any targets to succeed Conte, but suggested that the returning players could prove effective under a new leadership style.