Adkins led Saints to two successive promotions and had guided the club out of the Premier League relegation zone recently, but he was fired nine days ago and replaced by Mauricio Pochettino.
McDermott has been thankful that Reading owner Anton Zingarevich kept faith with him through the club's rocky period and criticised Cortese's treatment of Adkins, whose name was absent from Southampton's matchday programme for Pochettino's first game in charge.
"What I thought was sad was what Nigel Adkins has done for Southampton wasn't even mentioned in that programme at the Everton game. Let's have a bit of respect. That's not right," the Reading manager told a press conference after Saturday's win over Sheffield United.
"That's not right and proper. What he has done for that club has been phenomenal."
McDermott has grown close to Adkins over the last few years and he hopes the former Scunthorpe boss gets another shot at the big time soon.
"Nigel has earned the right to do so," McDermott added.
Reading took a break from their relegation battle to inflict a 4-0 defeat on Sheffield United in the fourth round of the FA Cup.
"There's a buzz around the place at the moment and that comes from winning games," McDermott said.