In the absence of the injured captain Kevin De Bruyne, Romelu Lukaku was handed the armband by manager Domenico Tedesco.
Courtois is one of the vice-captains of the Belgium national team, having won 102 caps since making his debut in 2011.
The 31-year-old was supposed to take the armband against Estonia on Tuesday but reportedly left camp after being disappointed with the captaincy snub against the Austrians.
“Together we decided that Romelu would be captain against Austria and Thibaut tomorrow against Estonia,” Tedesco said in a press conference.
“That was OK for everyone but after the match he [Courtois] suddenly wanted to talk to me and said he was going home because he was disappointed and felt offended.
“From the beginning I tried to show him the appreciation he deserves. In my eyes he is the best goalkeeper in the world. I love him as a goalkeeper, but also as a human being. I am shocked.”
Courtois' father earlier said his son pulled out of the squad due to a knee injury but Tedesco refuted those claims.
“I wish I could say it’s an injury but I can’t lie,” Tedesco said. “I always try to protect players but that’s impossible in this situation. I tried to tell him to hold on for two more days.”
