Sterling insisted afterwards that the confrontation, which took place in a private team area of St George's Park, was quickly forgotten about and both players were seen training on Tuesday.
Details of the clash initially appeared in the media shortly before England released an official statement, while specifics about the nature of the confrontation have since continued to emerge.
Southgate suggested he would prefer such information did not find its way out of closed camps and into the public.
"There seems to be all sort of information being passed from so many different areas," he told reporters.
"This is a consequence of the England football team at the moment. That is a very difficult situation and something we have to think about getting forward.
"The team getting out, information getting out – there seems to be so many voices in the background. That's something I have to accept and deal with. Every player has had an opportunity to speak to me if they want to and my view was the group is ready to move forward."
During the World Cup, an apparent England team sheet was leaked ahead of the group game against Panama, after assistant coach Steve Holland was photographed during an open training session with notes referring to a possible line-up.
At the time, Southgate was not hugely concerned by the information on the notes, although he did suggest the decision to publish the photograph was potentially to his team's detriment.
"It doesn't bother me in the slightest," he said in Nizhny Novgorod. "There's a squad of 23 names on the sheet, the next sheet has different players in different positions because we swap people in and swap people out.
"For me, no drama but obviously any time if we were to give the opposition the opportunity of having our team is a disadvantage to us. So, of course, our media has to decide whether they want to help the team or not. But, given that was just a squad list, it doesn't make any difference really."
Defender Ben Chilwell, meanwhile, suggested such leaks are to be expected in modern football – although he admits it is key to have the "full trust" of all players and staff.
"We're a very tight squad but it's not just the team in there," he said on Tuesday. "There's a lot of people who work at the [St George's Park Hilton] hotel involved as well.
"It's not just an England team thing, it happens in club football as well. Stuff does always seem to come out.
"As long as we have full trust that the stuff that needs to stay confidential between the team does then we're happy."