The pundit was discussing Griffiths' return to action and a gesture made towards himself when the Celtic striker scored in his side's 3-1 win over Kilmarnock.
He claimed Griffiths' professionalism was why he hadn't been playing as many games over the last few seasons.
Record Sport exclusively revealed Celtic had complained to Sky Sports while fans have contacted Ofcom over his comments, which included discussing his mental strength in deal with criticism from pundits and supporters.
And many felt Boyd had gone too far and was wrong to do it while having his own mental health charity.
Boyd set the organisation up after his own brother tragically died in 2016.
But the Rangers legend has slammed suggestions he was being critical of Griffiths over his struggles.
In a statement, he said: “My brother took his own life because of his mental health issues. The pain and suffering of his suicide still haunts me and my family each and every day.
"So please don’t accuse me of belittling another man’s mental health issues like I couldn’t care less. Just to be perfectly clear, on Sky Sports on Wednesday other night I mentioned Leigh’s mental STRENGTH in coping with criticism. I wasn’t talking about his mental health
“They’re two totally different things. So if anyone genuinely believes I’d have a go at Leigh Griffiths for problems he’s gone through then, I’m sorry, they are wrong.”