"I'm over it. I've left what happened at the ground and I've moved on," Archer wrote in his column for the Daily Mail.
"I don't want to go into the details of what was said but I know what I heard," Archer wrote.
"I thought members of the crowd around the guy might have pulled him up because I could hear him from the pitch as I was walking off," the seamer wrote in the column.
Archer had said earlier that he was subjected to "racial insults" by a spectator on the fifth day of the first Test against New Zealand at the Bay Oval on Monday.
"A bit disturbing hearing racial insults today whilst battling to help save my team," Archer had tweeted.
He added: "The crowd has been amazing this week except for that one guy. The Barmy Army was good as usual."
England lost the first Test of the two-match series against the Black Caps by an innings and 65 runs.
Meanwhile, New Zealand Cricket has said that it will contact and apologise to Archer.
"I know I wasn't hearing stuff. I told the security guard what had happened and that was it. Now my only goal is to make sure we finish this series on a high because we were all disappointed with the result in the first Test.
"But I found the incident a real shame. When you come to another country, you half expect fans to have a go at your cricket. If someone wants to shout at me and tell me I'm bowling badly, that's fine. I may not agree but it's fine. It's part of the experience of being a touring cricketer," Archer said.
"To hear racism, though that's another matter. There is no time or place for it in any walk of life, let alone cricket. It's just not called for," he added.
England will now aim to level the series when they meet Kane Williamson's men in the second and final Test beginning Friday at the Seddon Park in Hamilton.
( With inputs from IANS )