Players involved in the incident with Morpeth Town, following Newton's 3-2 FA Vase defeat, have also left the club.
Dixon was dismissed "due to the on-going ill-discipline" of some players.
"It's not been easy but we've acted straight away and want to draw a line under it," chairman Allan Oliver told BBC Sport.
"If there are any investigations we will fully take part, but it is difficult to comment further."
Newton Aycliffe, of Northern Football League Division One - the ninth tier of English football - have previously been the subject of a Durham Football Association investigation after Spennymoor Town refused to play the second half of a pre-season friendly in July after the match took an "aggressive direction".
Oliver said Dixon's departure was related only to Saturday's incident.
Morpeth's joint-chairman Gary Munday said the club, who play in the same league as Newton Aycliffe, are in the process of submitting a full statement to the Football Association as the incident is understood to have also been witnessed by the match referee.
While the club have not made a complaint to police, they said that if any player was to report an assault they will fully support any investigation.
Video footage of the clash at Newton Aycliffe's Moore Lane Park ground against their north east rivals shows a number of players from the home side running towards and gathering at the visitors' dressing room door.
Morpeth staff and coaches are said to have kept Newton Aycliffe players back, with at least one member of the home club's staff also trying to keep players out.
Oliver and another staff member are also understood to have walked Morpeth players back to their cars after the match.
Munday said he "had never witnessed anything like it at a football match" and "applauded Newton Aycliffe's strong stance" in dealing with the matter internally.
However, he fears it could have an effect on the club's support.
"We are shocked and disgusted by what happened," he said.
"Morpeth Town is a very family-orientated football club as we encourage our juniors, free of charge to support the club - that grass-root support is the future of football.
"Unfortunately, we have had parents say that they will never take their children to away games again.
"We will try to encourage them back as we do feel that this is a one-off."
The BBC have contacted the FA for comment, but are yet to get a response.

The motorcyclist, a woman in her 50s, and the driver of a Fiat Doblo crashed at traffic lights at the junction in The Avenue, Fareham at about 13:00 BST.
She was pronounced dead at the scene. The woman driving the Fiat was not seriously injured.
The eastbound carriageway was closed for several hours with drivers urged to follow diversions. The westbound carriageway remained open.

The Horizons project has already helped two dozen acts since 2014 - with radio airplay and UK festival appearances.
"Supporting new music is at the heart of what the BBC does across the whole of the UK, and internationally," said BBC Music's Jason Carter.
Bands and solo artists have until 12 February to apply.
"We're proud to support Horizons for a further year and see another 12 artists have a creative and potentially career changing 2016," said the Arts Council's Lisa Matthews.
"There's been some incredible opportunities created so far and we're looking forward with excitement to more incredible music experiences."
Solo artist Violet Skies said the 2015 project had meant a "mad year" for her.
She said: "I've been lucky enough to tick off a lot of things from my musical bucket list - The Great Escape, Festival No.6, SÅµn, Maida Vale and now, I'm looking forward to Eurosonic and SXSW festival in 2016.
"It's been a lovely little Welsh family of talented musicians and a really good support team - it's nice to turn up at festivals or a show knowing you have people there to help you."
Newton Aycliffe sacked their manager Peter Dixon after players from the non-league club fought their way into their opponents' dressing room.
[NEXT_CONCEPT]
A motorcyclist has died in a crash with a car in Hampshire.
[NEXT_CONCEPT]
A scheme to help new Welsh bands get wider exposure is set to return for a third year, backed by BBC Wales and the Arts Council of Wales.