The 33-year-old was due to fly out to the USA with the rest of the squad on Thursday but the club announced that the defender would not be travelling.
It is understood Koscielny, whose contract runs until next year, made it clear to the club that he wished to be released this summer in order to sign for another club.
But Arsenal stood firm and refused to grant the former France international a free transfer.
Wright is confounded at the timing of the furore surrounding Koscielny as Arsenal aim to challenge for a top-four place when the Premier League gets under way on August 9.
Speaking on his own YouTube channel, Wright said: “He’s given us unbelievable service, we’ve seen him play through pain and build a legacy and legendary status he deserves.
“It’s the time in the season where everybody is meant to be together. We are in for a tough season, a transitional period, we need a lot of catching up.
“For our captain to refuse to go on a pre-season tour after nine years of unbelievable service, to destroy your legacy a little by refusing to play for a club is something that is baffling me.
“Why Laurent? Who are your advisers and what are they saying to you?
“I am so disappointed in your decision to do this at this time, if your agent sees this, and I hope they do, shame on you, shame on you, for doing this to him and convincing him. No way you can tell me he will come up with this on his own.
“You’re the one we chose as our captain, our leader, and now we’re going into pre-season with our captain refusing to go on the tour.
“This is something I cannot get my head around and I pray to god it can be resolved sooner rather than later.
“It doesn’t set a good example, Arsenal need to do something quick and disciplinary. I love you Laurent but this is a poor decision.”