Napoli coach Antonio Conte said on Saturday that Kvaratskhelia had asked to leave the Serie A club immediately.
"It's hard for me, but it's time to say goodbye," Kvaratskhelia said in a video posted on Instagram.
"I spent an amazing time here, we shared a lot of memories together, experienced a lot of amazing emotions. Naples was my home, where I felt wonderful thanks to each of you."
Kvaratskhelia signed for Napoli in 2022 and was named Serie A's player of the season in the 2022/23 campaign after helping the club win their first Scudetto in 33 years with 12 goals and 13 assists in 34 games.
"Naples is a football city, Naples lives with football, and I am happy to have been a part of this great history," Kvaratskhelia continued.
"I learned a lot here, both on and off the pitch. I grew as a person and as a player... I would like to thank everyone who works at the club: the coaches, each member of staff, my teammates and the fans.
"I say goodbye, but you will always be in my heart, and I hope we will meet again. I know that your heart is very broken, but one day I will tell you everything."
The Georgia international had previously been linked with a move to PSG in the close season before he stayed under new coach Conte.
His Napoli contract ran until 2027.