Williams had spent more than a decade at the club, as a player, assistant boss and then manager following Phil Parkinson's departure in July 2006.
Coyne is saddened by the 46-year-old's departure, telling The Gazette: "The manager is the first person to get it in the neck and the disillusionment towards Geraint on Saturday was a big factor in him going.
"It was disappointing to see that directed towards someone who had been such a great servant for the club for so long and a great person.
"I can understand why the fans were unhappy but it wasn't nice to see.
"The atmosphere in the stadium was more noticeable in the last five minutes when we were pushing forward and conceded the two late goals."