Wright was angered by the nature of Koscielny's social media video unveiling at Bordeaux.
In the clip, Koscielny, who left Emirates Stadium after nine years on Tuesday, pulls off an Arsenal shirt to reveal his new team's top underneath.
The 33-year-old defender refused to travel on Arsenal's pre-season tour and made it clear he wanted to leave.
Wright said Koscielny's actions "hurt".
Directly addressing former France centre-back Koscielny, ex-England striker Wright posted on social media: "You should be ashamed for the way you've left the club after nine years.
"Got what you wanted and still trying to have a dig. Hope it's worth it in the long run."