Celtic captain Brown, who was persuaded out of international retirement by Strachan, posted a photo of the outgoing Scotland boss on his Instagram page with the caption: “Sad sad sad day.
“We all had faith in Gordon and believed in making the Euros. 14 points out of 18 in 2017 momentum was on our side.”
Brown, who suggested earlier this week that he would carry on playing for Scotland if Strachan stayed on, initially stepped back from the international scene in mid-August, before returning barely two months later.
He said in August: “It’s no secret that I’ve suffered some injuries in my career and once you get over 30 you have to protect your body more.
“I’m not the type of player who can coast half-heartedly through a game - I give 100 per cent every time.
“Given the importance of this campaign for Scotland, and the challenges ahead at Celtic, I simply felt that I could not keep up both commitments without either my performances or recovery time suffering.”