The club boss responded on Twitter after a series of social media accounts expressed outrage about the footage, which shows fans on a bus singing a song which references the late former Newcastle United, Leeds United, Everton, Sheffield United and Bolton Wanderers midfielder and former Wales manager.
One fan seen in the centre of the video, has been singled out for criticism, with calls for him to be banned from matches.
In a reply on social media to a tweet asking if he was frightened to condemn the fan, Mr Donald replied: “No I’m not - it is shocking.
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“I have seen it and I absolutely gutted.
“It is unacceptable, unjustifiable, wrong, hurtful and embarrassing.
“Gary Speed’s poor family deserve support and understanding and to abuse such a tragic event is simply wrong and sick.
“I’m sorry and I’m on it.”
His message has been met with support from some, who have said they were pleased to see the matter was being dealt with.
Fans’ favourite Speed, who was awarded the MBE in 2010, died in November 2011.
His death rocked the footballing world.
During his illustrious career he won 85 caps for Wales.
While at Leeds he won the League Championship in 1992 and while at Newcastle United he was part of sides that finished runners up in the FA Cup in 1998 and 1999.