"You're not supposed to take centre stage if you are a physician to a team. You have to understand your dynamic in the club. You are not a player,' Rogers told The Daily Mail.
"Her putting that "thanks for the support" on Facebook was extremely naive. That would upset anyone. What was she trying to achieve? Are you bigger than the manager? You're never going to win, nor should you.
"You're the backroom staff, you're there to do a job. You're not there as a cheerleader. You can play to it or you can play it down. It seems like she played to it. You're not supposed to be a celebrity. You can't be a celebrity doctor, come on. Let's get real."
Carneiro has retained her job title, but with severe restrictions. She cannot attend matches, training sessions or the team hotel.
Rogers resigned at Chelsea in 2011 after the appointment of Paco Biosca as medical director.
He was a senior member of the backroom staff under Carlo Ancelotti and then Andre Villas-Boas and is now medical adviser to the NBA as well as being sports physician to Warwickshire County Cricket Club.
The departure of Rogers saw Carneiro eventually promoted through the ranks.
The Football Medical Association on Tuesday stressed that Carneiro "acted with integrity and professionalism at all times" but Rogers believes her demotion is down to other engrained difficulties between medics and management.
"Who's to say when common sense prevailed that everything would have settled down,"he added. "But when you add the Facebook and continue with it because you are a "celebrity doctor" this is what happens.
"There is no chance she is still the first-team doctor. You can't do your job. There are two sides to every story - especially because I've heard him praise medical staff in the past.
"There's something else going on. If you had a rapport with someone that important in your club you wouldn't have a go like that. The medical team need to have a relationship with the manager. If you have respect, that kind of thing never happens.
"With a head injury you run on, period. But you have to understand the injury and the game. I completely agree with some of the things Mourinho said."
Mourinho has had serious misgivings about the fitness of £32 million striker Diego Costa since his arrival from Atletico Madrid last year.
Rogers said: "Who is to say that serious medical issues didn't happen in the past and this is the straw that broke the camel's back? My feeling is that he was frustrated and those frustrations were deep-seated. They weren't just, "Oh you ran on the pitch, you made a mistake".
"His reaction had nothing to do with that mistake. Don't you think that was an over-reaction? He could've deflected onto something else."
