The award-winning host also criticised the ABC for failing to support him publicly while he was subjected to the abuse.
On Sunday, Anderson sent an email to ABC staff apologising to the Wiradjuri man.
“How the ABC supports people in these moments is important,” he said.
“Stan Grant has stated that he has not felt publicly supported. For this, I apologise to Stan.
“The ABC endeavours to support its staff in the unfortunate moments when there is external abuse directed at them.”
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Anderson also hit out at “some commercial media outlets” for what he described as “sustained and vitriolic” “anti-ABC reporting”, saying it had real-world effects on presenters and journalists.
Anderson described Grant’s experiences since the coronation broadcast as “distressing and confronting”.
“Racism must never be tolerated, and I am dismayed that Stan has been exposed to such sickening behaviour,” Anderson said.
“Stan has our full support. And he has always had our full support. Stan makes an enormous contribution to conversations of national importance.”
The ABC’s Bonner Committee, which advises the broadcaster on matters linked to Aboriginal and Torres Strait Islander staff, content and other issues, asked the managing director to review the way the organisation responds to racism affecting staff, a request he accepted.