The Scotland international defender signed a five-year deal with the Gunners on the English Premier League's transfer deadline day after 15 years at the Scottish champions.
Tierney, who made his senior debut for the Hoops in April 2015, won eight trophies with Celtic and was part of the two treble-winning sides. He made the transition from fan, to ballboy, to youth player to occasional captain.
Writing on Instagram and Twitter, alongside a series of photos of him in Celtic gear, Tierney said: "This is coming from the bottom of my heart.
"I don't know how to even begin to put the last 15 years of my life into words... what you, as fans, have given me and my family I cannot repay you in any way possible and for that I am sorry.
"From the age of seven, I was lucky enough to play for the club I love and will continue to love until the day I die no matter what happens.
"Every single second of my Celtic career was a dream and I gave mty all every single time I stepped onto that pitch - it's honestly the least I could do for everyone involved at the club."
Thanking the coaches and staff members who helped him on his rise to the top and paying tributes to his "legendary" team-mates, Tierney added: "Where can I start with the fans?
"I tried my best to represent you every time I went onto that pitch. The relationship we had together and the memories we created and history we made was incredible.
"I will cherish every single second of it. I hope we can continue to have a relationship although I understand that will be never for some people, which is heartbreaking for me, although I understand the emotions surrounding the situation.
"It has been the hardest day or two in my life and I really appreciate every single message of support. This was not easy in any way, shape or form.
"To end I'd like to mention the special relationship I had with the ultras who I have grown up with from BHOYS and the Green Brigade.
"The atmosphere you create to help the team is massive and it helped me so much over the last few years. I thank you so much for these times together - what we have achieved as a club in the last few years isn't possible without you and all the fans.
"This is not nearly enough of what I am feeling but I literally cannot put into words the emotion I'm feeling towards you all.
"I hope I showed you every game that I was lucky enough to play. God bless."