Mourinho has been cleared of making discriminatory comments to Carneiro, who has since left the club.
But Dyke, in a letter to FA Council members, said the 52-year-old Portuguese "made a mistake".
The FA has been criticised by one of its board members - Heather Rabbatts - for its handling of the case.
Rabbatts, head of the FA's inclusion advisory board, said the governing body's reaction had been "seriously disappointing".
"I have major concerns over the way in which the disciplinary process has been conducted," she added.
The FA investigated after a member of public made a complaint about Mourinho's language towards Carneiro during Chelsea's 2-2 draw with Swansea on 8 August.
Carneiro and head physio Jon Fearn had treated Eden Hazard for an injury after being called onto the pitch by the referee, which meant the player had to leave the field.
That meant Chelsea were reduced to nine men, with goalkeeper Thibaut Courtois having been sent off, and Mourinho criticised the pair in a post-match news conference.
Carneiro, 42, had her role downgraded before deciding to leave the club.
"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," said FA chairman Dyke.
"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 judgement and public behaviour."
The Women in Football campaign group said its language expert is certain Mourinho used abusive language, contrary to the verdict of the FA's chosen expert.
But the FA said it had appointed an independent academic expert in Portuguese linguistics to study the footage of the incident, which included the audio recording.
"We have engaged an expert of high calibre to address such an important issue such as discriminatory language, which we take very seriously," said a spokesperson.
Rabbatts said the FA had failed to act previously over sexist and abusive chanting towards Carneiro by fans in stadiums.

After taking part at London 2012, the four national football associations could not agree on sending a British women's team to the Rio Games.
A place was earned by England finishing third in the 2015 World Cup.
"To turn it down is a great shame. It's something we may look at in the future and try to compete in," Allardyce told BBC 5 live's Sportsweek.
"When you see the delight on Justin Rose's face after winning the gold medal in golf it shows what it all means."
England's Football Association had put forward the idea of sending Great Britain teams to the Olympics, but Fifa said it would need the agreement of the ruling bodies in Scotland, Wales and Northern Ireland, who were against it.
The other home nations fear such a move could affect their independent status within the sport's world governing body, Fifa, and at international tournaments such as the World Cup.
Team GB entered men's and women's sides at London 2012, with Stuart Pearce coaching the former and Hope Powell the latter. Both teams were knocked out in the quarter-finals.
For European nations, qualification for the Olympic football competition is via the Uefa Under-21 Championships for men and the World Cup for women.
England were the only home nation to compete at the Under-21 Championships, but failed to secure the required semi-final berth.
British Olympic Association vice-chairman and former sports minister Sir Hugh Robertson believes politics in particular has denied the women's game a perfect platform to build on the interest it received at London 2012.
A total of 154,998 people turned out to watch the four Team GB women's matches four years ago.
"From the British Olympic Committee's perspective, we would love to see Team GB football," Robertson told Sportsweek.
"It is particularly a powerful tool to promote the women's game.
"The tragedy is that the politics of football administrators impact on the athletes because women football players would want to be at the Olympics."
Chelsea manager Jose Mourinho should have apologised to ex-club doctor Eva Carneiro for his criticism of her, says Football Association boss Greg Dyke.
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England manager Sam Allardyce would welcome Team GB entering football teams at future Olympics.