In a letter to FA council members, Dyke says Mourinho should have apologised.
The letter was sent following Wednesday’s FA board meeting, since when independent board member Heather Rabbatts has launched a stinging attack on the FA for its “seriously disappointing” handling of the Carneiro case.
Dyke’s letter makes clear his misgivings over Mourinho’s behaviour.
Carneiro was dropped from first-team duties after an incident on the opening day of the season when she was criticised by Mourinho for going on to the pitch to treat Eden Hazard. She has since parted company with Chelsea.
Dyke says in the letter: “There have been some well-documented issues of late around equality and inclusion in the game, an issue where it is vital we continue to show clear leadership.
“I felt the handling of the case of the Chelsea doctor, Eva Carneiro, was a good example of this. We supported Heather Rabbatts’ strong statement on the matter.
“Personally I don’t think Mr Mourinho comes well out of the whole saga — he clearly made a mistake in the heat of a game, and should have said so and apologised.
“Instead he has said very little and Miss Carneiro has lost her job.
“Our regulatory team have investigated this and whilst Mr Mourinho has breached no rules it was clearly a failure of his personal judgment and public behaviour. This should be seen as such by the game.”
Mourinho was cleared by the FA on Wednesday of making discriminatory comments towards Carneiro. It has emerged the FA did not ask to interview Carneiro, although it is understood it asked her lawyers if they wished to provide any evidence.
Meanwhile, Southampton manager Ronald Koeman will look to exploit any lack of confidence in struggling Chelsea’s back line when his team take on the Premier League champions at Stamford Bridge tomorrow.
Chelsea are 14th in the table having already conceded 14 goals in seven games.
“Maybe they have some defensive problems,” Koeman said yesterday.
“Last season they were the team, along with our defenders, who had the best record in defence. They didn’t concede a lot of goals.
“This season it is different. Maybe the confidence is not quite the same.”