The sandstone war memorial opened in 1941 to commemorate Australians killed in World War I and is among Canberra's most popular tourist attractions.
Nelson had proposed replacing the phrase "known unto God," attributed to British writer Rudyard Kipling, with the inscription: "We do not know this Australian's name, we never will."
Those words open a eulogy given by then-Prime Minister Paul Keating for an unknown soldier killed in WWI, exhumed from a French cemetery and re-interred at the memorial in 1993.
Keating was a polarizing politician who led the center-left Labor Party. Abbott leads the conservative Liberal Party and Nelson is a former Liberal leader.
Nelson said some complainants "had particular views about Mr. Keating." Others accused Nelson of "de-Christianizing" the memorial, which he said was always intended to be a secular institution.
"This was never driven by some suggestion that we should remove 'God' or political correctness or anything of the sort," Nelson said. "The motive was to give permanence to this towering Australian speech by an Australian prime minister."
The memorial's governing council has settled on a compromise that will include Keating's 1993 words "He is all of them, and he is one of us" being inscribed in the stone surrounding the soldier's grave.
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Online: Australian War Memorial site: http://www.awm.gov.au/