Cooper's withdrawal from the role comes four years after the sprint specialist was sent home from a training camp for the Games in Birmingham for misuse of prescription medicine and wellbeing issues.
Cooper, who helped Australia win the 4x100m medley gold medal at Glasgow, had failed to meet team expectations and had stood down from the team's "Athlete Leadership Group", Swimming Australia (SA) said in a statement.
"We look forward to watching Isaac compete at this month's Pan Pacific Championships."
Cooper, 22, was not competing on the final night of the swimming in Glasgow, but Australia's management expect their athletes to support teammates by attending.
The Sydney Morning Herald newspaper reported Cooper was out drinking in Glasgow on the final night of competition, citing sources with knowledge of the situation.
SA declined to comment.
Cooper and his management did not provide immediate comment when contacted by Reuters on Tuesday.
Cooper won the 50m backstroke gold medal at the 2024 world championships in Doha and earned Olympic bronze at the 2020 Tokyo Games as a 17-year-old in Australia's mixed medley relay team.