Villa moved out of the bottom three of the Premier League with a goalless draw against West Ham United, but the main talking point to come from the game was a clash between the Villa team-mates.
The incident came late in the first half when Mings and El Ghazi clashed angrily inside their own penalty area, with the latter clearly pushing his head into his team-mate's.
El Ghazi could have been sent off but, with referee Mike Dean receiving instruction from the video assistant referee, escaped a red card.
Smith was unmoved by what happened and was more concerned that his team failed to make the most of Arthur Masuaku's dismissal in the 67th minute, with West Ham the more threatening despite their numerical disadvantage.
"I didn't need to deal with it, the players dealt with it," said Smith. "It was dealt with by the players in the dressing room and I'm happy for that to happen, that's the spirit in the dressing room.
"Things happen in team sport in the heat of the moment and it was sorted very quickly. At half-time they spoke about it, shook hands and we move on."
Smith was more bothered about the fact his side could not find a way through the weakened Hammers team.
Masuaku, booked in the first half, received a second yellow card for a soft-looking foul on Villa substitute Ahmed Elmohamady.
Thereafter, Villa failed to really worry West Ham, instead inviting pressure on themselves through individual errors and silly fouls inside their own final third.
"What is disappointing is that we played just over 20 minutes against 10 men and couldn't take advantage of that," he added.
"We lost our structure and made mistakes and at times lost emotional control during that last 20 minutes or so."