Joyce confronted the coach after the game, which led to a heated exchange.
Joyce said yesterday he would lay a complaint with Football South regarding the incident.
Until then, he would not name the team or coach so "due process" could be followed.
He did not know why the coach made the child rub his nose into the ground, he said.
"One of the parents said it was something the coach did to make them look at their feet, and he wasn't looking at his feet."
The coach bragged like a "rooster going around a farmyard" to parents following the incident, Mr Joyce said.
"That guy's obviously going to have to justify himself at some time."
Football South chief executive Chris Wright said he had been alerted to the incident.
He received information, both from people defending the coach and from those expressing upset at the behaviour, he said.
He would not name the club, but said once he had received more information, the organisation would follow processes that involved speaking to the coach.
"Like many sports, we take the quality of experience of the players very seriously.
"But also, the information we have at the moment is quite vague, so we need to gather more information."
Until Joyce contacted him, it was hard to comment, he said.
"I think it's really important that it's not a witch hunt. While on the face of it this sounds concerning, and certainly something we need to look into, we don't want to make any judgements at this stage."
