Clymping, who were trailing 5-0 at the time, had three players sent-off in the game and manager Darran Atkins has since left the club.
Clymping released a statement on Sunday, in which chairman Dean Adams said: “What happened yesterday during our game with Southwick has left our club deeply saddened and shocked.
“We are embarrassed and stunned by the player’s actions, which I am led to believe is totally out of character.
“We cannot apologise for him, and we cannot stop individual’s actions – but we can make sure a repeat of this never happens again.
“The player concerned will never play for this club again, if he is ever allowed to play football in the future.
“As a result of yesterday, both the club and the manager’s reputation will get tarnished over something they couldn’t control.
“Both the manager and the club are angry and upset by what has happened.
“I have personally spoken to Phil Wilkes, the referee involved yesterday, and apologised on behalf of the club. What was said will remain between us, but I don’t think he will mind me saying that ‘idiots’ will not put him off his job.
“I’d like to apologise to Southwick and all those involved or watching yesterday, it was totally unacceptable.
“Lessons need to be learned from this, as verbal and personal attacks will one day kill the game at this level due to a shortage of referees.
“We fully support any campaign to support referees and this, as chairman, is something I will look into this week.
“Again, apologies to everyone in Sussex football for this incident.
“As a result of our difficult season, the club have decided to make a fresh start ahead of our last few games, and also for next season.
“The club feel it is now in their best interests to part company with our manager, Darran Atkins, with immediate effect. Darran was let down by certain people yesterday and tried to check on the referee, while also apologising to Southwick post-match.
“Darran joined the club in difficult circumstances and helped us during what is a forgettable season. Darran is a passionate fella and we wish him all the best in his future in football. I would like to make it clear he is in no way responsible, nor we as a club, for individual actions beyond our control regarding yesterday.”