"We also confirm that SA Rugby and Erasmus have advised World Rugby that they withdraw their Notice of Appeal and will not lodge an appeal against the sanctions imposed by the Judicial Committee.
"This has been a highly stressful and charged environment with unusual pressures placed on all concerned and we have no wish to prolong that experience for anyone.
"We have drawn a line under the incident and only wish to look forward. We will respect the outcomes of the hearing, allowing our national teams and rugby operations to plan with clarity for the coming months."
World Rugby also released a statement saying: "World Rugby welcomes the public apology from SA Rugby and Rassie Erasmus to the match officials involved in the first test between South Africa and the British and Irish Lions this year and the matter is closed."
Erasmus, as a result of his suspension, missed last weekend's fixture between England and South Africa at Twickenham, with the world champions narrowly losing 27-26.
The six charges brought against Erasmus included threatening a match official that, unless a requested meeting took place, he would publish footage containing clips criticising the match official's performance and then making good on that threat, and for also engaging in conduct or activity that may impair public confidence in the integrity and good character of match officials and for bringing the game into disrepute.
SA Rugby, meanwhile, were sanctioned for not ensuring that Erasmus "complied with the World Rugby Code of Conduct and/or permitted Mr Erasmus to commit acts of misconduct", along with not preventing captain Siya Kolisi and assistant coach Mzwandile Stick from making "comments at a press conference on July 30, 2021 that were not disciplined or sporting and adversely affected the game of rugby".
The withdrawal of the appeals and apologies from Erasmus and SA Rugby welcomingly bring one of the sport's uglier off-field moments in recent years to a close.