In addition, Erasmus and the Springboks' union have issued a public apology to the match officials, led by referee Nic Berry, who oversaw the first Test against the Lions in July.
Erasmus was banned from all rugby activity for two months and can not be involved with any team on matchday until September 30 next year after being found guilty of six misconduct charges.
Among the proven accusations against South Africa' s director of rugby was that he threatened Berry in the hope of setting up a meeting with the Australian. He then released on social media an hour-long critique of the match officials' performance.
SA Rugby, meanwhile, was fined £20,000 and warned over its future conduct after two counts of misconduct were upheld.
Both Erasmus and the union were also ordered to apologise to the match officials, which has now been made public.
"SA Rugby and Rassie Erasmus wish to apologise to the match officials appointed to the first Test of the Springboks' series against the Lions," an SA Rugby statement read.
"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."
Following the verdict of the hearing, Erasmus received strong support in South Africa despite the extent of the emotional toll taken on Berry coming to light in the written judgement.
Last week the Springboks boss appeared to make light of the sanctions by posting on social media a photo of him in camouflage, suggesting that he might attempt to gain entry to Twickenham for the England game.
And in another slight on World Rugby's disciplinary process, he published a photo of a pint of Guinness with a caption saying "I do however get the feeling the 2 months will not be so bad as feared!!".