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Steve Clarke's side slipped to a 3-1 defeat at Hampden Park and Ukraine will now face Wales on Sunday for a place in Qatar next winter.
Liverpool left-back Robertson was spotted at his side's FA and Carabao Cup trophy parade over the weekend and appeared to be holding a beer in his hand.
Supporters were furious at the time and after Robertson put in a below-par performance against Ukraine, they've completely had enough.
A number of tweets on social media want the 28-year-old punished, after seeing the Tartan Army fail to reach their first World Cup in 24 years.
One commented: "Andy Robertson should be stripped of the Scotland captaincy. Serves up a performance like that? An absolute disgrace of a performance both on the park and off the park."
Another said: "Here at least Andrew ‘Andy’ Robertson got a bevvy after winning a diddy cup double eh."
A third put: "Callum McGregor the captain and leader that Andy Robertson thinks he is."
While someone added: "Get that captaincy taken off Andy Robertson."
Scotland boss Clarke had backed his player heading into the game and thinks it was the 'right thing to do' after a tough week professionally.
Liverpool were pipped to the Premier League title by Manchester City and then lost the Champions League final to Real Madrid.
"I saw the clip and I was pleased he was getting over his disappointment. Some people want to make too much out of small things," Clarke declared.
"It was the right thing to do, the right thing to celebrate with his team and his supporters on the back of a fantastic season for Liverpool.
"He has joined us in a great place and that’s all I am concerned about.I don’t care about criticism. It won’t bother Andy and it certainly won’t bother me."
Andriy Yarmolenko put Ukraine ahead in the first half before Roman Yaremchuk doubled their advantage just after the break.
Callum McGregor halved the deficit with 11 minutes to go to set up a gripping finale.
Scotland failed to find an equaliser however and Artem Dovbyk scored a third late into injury time.