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.
Chelsea manager Jose Mourinho should have apologised to ex-club doctor Eva Carneiro for his criticism of her, says Football Association boss Greg Dyke.