The tireless flanker had been named to lead the side in his 122nd Test on Sunday morning (AEST) but will now return to Australia, missing both matches against the Pumas.
Fellow Test centurion James Slipper will captain the side in Mendoza, while Fraser McReight will replace Hooper in the number seven jersey.
"While this decision did not come easily I know it is the right one for me and the team at this point in time," Hooper said in a statement.
"My whole career I've looked to put the team first and I don't feel I am able to fulfill my responsibilities at the moment in my current mindset."
Coach Dave Rennie said his captain, aged 30, had shown "true courage".
"Michael's one of the most professional and impressive men I've coached I know this has been a difficult decision for him," he said.
"He's shown true courage by acknowledging where he is at and acting on it.
"We will support him in any way we can and I know the team will be focused on getting the job done tomorrow."
Hooper captained Australia during their recent 2-1 series loss to England at home and has been a regular presence in the side since making his debut in 2012.
The Wallabies, who have a World Cup in France next year, return to Australia for Rugby Championship Tests against South Africa and New Zealand beginning later this month.
"Michael is an incredible leader, it takes a brave man to identify where he's at and come forward whilst having the best interests of the team at heart," Rugby Australia boss Andy Marinos said.
"His wellbeing is and remains the highest priority right now where Rugby Australia and the Australian Rugby community will do everything to support him and his family."Â