The chair of the Council for the Order of Australia says honours will not be stripped from recipients unless the awardee has had a criminal conviction.
Ms Arndt tweeted congratulations to the Queensland police for "keeping an open mind" on the deaths, "including the possibility that Rowan Baxter might have been "driven too far"."
"Short of a criminal conviction, we are not going to be arbiters of what people can or can't say," Mr Stone said.
Men's rights activist Bettina Arndt was condemned in the Senate for her comments on the murders of a Queensland mother and three children.
Mr Stone's comments come as two new members of the council were appointed in the last week: Jillian Segal AO and Melinda O'Leary, replacing Elizabeth Broderick and Gabrielle Trainor, both of whom are understood to have opposed the award to Ms Arndt.
But Mr Stone says Ms Broderick's term had come to an end and she was fully occupied with her work on gender equality for the United Nations.
"Her UN work had her out of the country for most of the time and we need 100 per cent participation," he said.
Mr Stone said that he believed Ms Broderick had a lot to offer and would return to the council when her UN commitments were complete.
All new community appointments are made to the council through the Department of Prime Minister and Cabinet and Mr Stone says neither he nor the council has any input. He also says that he believes the entire process should be more open and transparent than it has been in the past.
"We need a broader discussion about how we operate and how the system works," he said.
He said that would not include commentary on individual recommendations of the council.
Mr Stone said that on the matter of gender equality in the council itself, he was very pleased that during his tenure as chair, the council had achieved an equal gender split.
Ruth McGowan, co-convenor of lobby group Honour A Woman, welcomes any move to greater transparency.
"We expect the community appointment process to reflect the diversity of the Australian community and follow Commonwealth guidelines, she said.
"We would like to see an overhaul of the whole process so that both appointments to the council and the assessment processes are more open and transparent."
Honour a Woman is calling for targets for diversity and all levels of Australian honours.
The council is made up of community representatives, representatives from the states and territories, ex-officio representatives.