"Stan has our full support. And he has always had our full support," Mr Anderson wrote in an email to ABC staff.
"Stan makes an enormous contribution to conversations of national importance."
Grant made the decision to leave Q&A last week after racism against him intensified following his involvement in the public broadcaster's coverage of the King's coronation.
The veteran correspondent was invited to participate in an ABC TV discussion on coronation day, during which he pointed out that the crown represents the invasion and theft of Aboriginal land.
In his email to ABC staff, Mr Anderson said he agreed with a recommendation from the ABC's Indigenous advisory committee to review how the broadcaster responds to racism towards staff.
He said anti-ABC reporting from some commercial media outlets had been "sustained and vitriolic".
"This has real-world consequences for ABC presenters and journalists who are personally attacked and vilified," he wrote.
"How the ABC supports people in these moments is important. 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."
Grant last week expressed frustration at the ABC's executives, saying not a single one had publicly refuted the lies that have been written and spoken about him.
"I don't hold any individual responsible; this is an institutional failure," he said.
While the ABC's news director Justin Stevens had been a "support and a comfort" according to Mr Grant, the national broadcaster had its own legacy of racism.
In a statement last week Mr Stevens said the broadcaster stood by Mr Grant and the abuse was "abhorrent and unacceptable".
Threats against Grant had been referred to police while a formal complaint had also been lodged with Twitter.
The ABC Ombudsman would investigate editorial complaints regarding the panel, Mr Stevens said.
