The former Parkhead favourite lambasted the 30-year-old – who has struggled with mental health issues in recent seasons – for failing to get himself properly fit for this season.
Griffiths climbed off the bench to net Celtic’s only goal in their humiliating 4-1 home Europa League loss to Sparta Prague on Thursday evening.
His introduction briefly threatened to spark a comeback from the home side before Lukáš Juliš completed his hat-trick to settle the contest. Substitute Ladislav Krejčí then added a fourth in injury-time.
Speaking on BT Sport’s live coverage after full-time, Sutton criticised the player for not being fit enough to start matches for the club and his former team-mate, manager Neil Lennon.
He said: “I am a huge, huge fan of Leigh Griffiths as a player. I am not a huge fan of his attitude.
“He showed what he can do. He is a brilliant, brilliant finisher. But he is overweight, out-of-shape and he can’t play for 60 minutes, let alone 90 minutes.
Asked if he could start for Celtic, Sutton responded: “Are you serious? Look at the shape of him! He’s a professional footballer. He shouldn’t be paid to come off the bench for a little cameo for half an hour.
“He’s a brilliant player, I rate him highly, but he needs to give himself fit. He’s let his team-mates and his manager down with his attitude.
“Can you see him starting, the shape he’s in?”