Garnacho stormed away to the home dressing room after being changed before halftime in victory over Ipswich Town last night. United manager Ruben Amorim made the change after Patrick Dorgu's red card.
"He's walking on thin ice, Garnacho, he is," Saunders told talkSPORT.
"If you look at the league table, there's not a player at Man United who can complain to the manager for not playing or getting brought off, look where they are in the league.
"And if he's, from what I've heard, he's arguing with the manager, he's arguing with the tactics and he's throwing his toys out the pram when he gets brought off.
"He's going to ruin his career if he carries on."
On his own experience managing, Saunders also said: "I used to say to the group, 'Listen, your job is to go out and do your job in your role, whether it's a striker, getting goals, making goals.
"My job is to pick the right team, give you some tactics with and without the ball, and then the only way I can affect the game is by making substitutions in the last half an hour.
"So if I bring you off, don't take it personal, and don't... There's people stood behind me, directors and owners of the club.
"When you're shaking your head walking off, how do you think I look standing in the technical area?
'Come and see me Monday morning, knock on my door and I'll explain to you why I took you off. Don't show it out there in front of everyone."