It is understood the Hawera High School student was given a warning after telling police it was from a fake account and the abuse had not been sent by him.
The formerly Durban-based family, who moved to New Zealand in July, immediately put their son on a bus to Auckland after the incident and followed him later for Christmas.
When they returned to Hawera on Tuesday, the victim's mother found a letter in their post box.
The typed note, claiming to be from the perpetrator and signed off as "John Doe", apologised for the offensive behaviour.
"I do not wish for the police to continue their investigation and my one want is for everybody to move [on] over this," they wrote.
But the victim's mother was not satisfied and carefully replaced the letter in the envelope.
She said the police were now fingerprinting it.
The writer said they had set up a fake Facebook account in response to the South African teen making threats to someone else earlier in the year - an allegation he denied.
"I never expected to go as low on the insults as I did. I am not a racist person, this is the lowest point of my life that I have ever been at," they wrote.
"I ruined your Christmas, that thought alone has kept me up at nights. Crying. Wondering how I can make this right."
The writer said they would seek counselling over the incident and promised not to do it again.
The mother of the recipient said she was "shocked" by the note and hoped the police found whoever was responsible. "I don't know what to make of it anymore," she said.