Ashley released his public apology following a day of lengthy talks with manager Alan Shearer and managing director Derek Llambias on Tuesday.
Ashley conceded that mistakes had been made during his 18-month reign on Tyneside which ended in Sunday's departure from the top-flight.
"Seeing Newcastle United relegated from the Premier League has been a catastrophe for us all," he said.
"I fully accept that mistakes were made during this and previous seasons and I am very sorry for that. I would like to say sorry to our magnificent supporters who have turned up in their thousands, home and away, up and down the country and who have given the team amazing backing throughout.
"And I would also say sorry to all of the club's hard-working staff, who have gone about their jobs all year long."
Middlesbrough will not stand in Stewart Downing's way should he want to move on following the club's relegation, according to chairman Steve Gibson.
Boro rejected several bids for the England winger from Tottenham Hotspur in January as they battled to stay in the Premier League.
Gibson also gave the manager Gareth Southgate his full backing, saying he would not be sacked.