The Owls have also revealed they will be investigating the circumstances of the 19-year-old's move to Parkhead under freedom of contract in the summer when his Hillsborough contract runs out.
The switch was announced by former Celtic chief Neil Lennon several weeks ago - to the disappointment of Wednesday.
Posting on Twitter, Shaw said: "First of all I would like to thank everyone at Sheffield Wednesday - the owner, all the staff, my teammates and especially the supporters.
"After discussing the options with my family I have decided to take on a new challenge and so, I’ve agreed to join Celtic, one of the most famous and prestigious clubs in the world on July 1.
"Sheffield Wednesday is a massive and brilliant club and I’m extremely grateful for everything the club has done for me.
"The club will always be special for me as I grew up a fan and nothing can change that for me.
"In the decision I based my decision purely on the football aspects. By joining Celtic next season I will fight for a place in a team challenging for titles, trophies and to play in Europe, which is something I’ve always dreamed of."
Confirming Shaw's departure north of the border at the end of the season, an Owls statement said: "We are currently investigating this matter and will consider if there are any further steps available to protect the club’s interest.
"It was not our intention to place this development into the public domain out of respect to all parties concerned.
"However, we were contacted by Celtic informing the club of their intention to formally confirm this development, which was revealed by the former Celtic manager in a media briefing.
"Both clubs exchanged statements in advance of a coordinated release but Celtic have since ceased correspondence to this end.
"Shaw remains a Sheffield Wednesday player until the expiry of his current contract and will be available for selection for the remainder of the season.
"We will make no further comment at this time."