Hazard lost patience in the Capital One Cup semi-final second leg match in south Wales as ball boy Charlie Morgan refused to hand over the ball after it had gone out of play for a goal-kick. Morgan fell to the ground as Hazard attempted to get the ball from him, with the Blues forward then trying to kick it from under him but appearing to instead make contact with the youngster.
Referee Chris Foy dismissed the Belgian international and Gallagher said on Thursday: "The referee has taken the correct action, It's against the laws of the game to commit violent conduct on anyone under the jurisdiction of the game. I can't reinforce how switched-on the officials were. It would have been so easy for the assistant referee not to pick that (incident) up - but he did."
Gallagher continued: "There's no defence for that. Someone said it was borne out of frustration but you have to accept it. I don't want to see it again.
"It was an extreme but you can't have that at a football match. It's been addressed - and the action which has been taken is strong and correct."
The Belgium international is now facing a three-game ban - and possibly further action from the Football Association.
Gallagher added: "I don't know what the FA will do. Chris will have reported it, he's done his job on the night. But it is an extraordinary incident and it's up to the FA to decide what to do."
The former official explained that referees can remove ball boys from matches if they believe there is a problem.
The 55-year-old, who retired from top flight action in 2007, explained on Sky News: "Chris had the power to go to the stewards and get a ball boy removed - and add on time.
"But the people who look after ball boys are very, very good. They are spoken to before each game by each individual referee, who will say 'if we have a problem, we can address it'.