The delegation has also been told Canberra expects fugitive Solomons attorney-general Julian Moti, an Australian citizen, to be handed over to Australian police to face child sex charges.
Members of the four-nation Melanesian Spearhead Group (MSG) met with officials from the Department of Foreign Affairs and Trade (DFAT) in Canberra to discuss their recent visit to the Solomons capital Honiara.
The group consists of Papua New Guinea, Fiji, Vanuatu and the Solomon Islands, but an official from Honiara was not among those who met with DFAT's Pacific division first assistant secretary David A Ritchie.
Officials from the department of Prime Minister and Cabinet and AusAid were also at the meeting.
DFAT said it raised the "extraordinary and unwarranted" expulsion of Canberra's ambassador from the Solomons, "deep concerns" about a commission of inquiry into the April riots in Honiara, and the need to continue the work of an Australia-led assistance mission.
"We also said that due process was expected to be followed in relation to Julian Moti," a DFAT spokeswoman said.
"Australia has made it clear that it has not sought and does not see the need for mediation in its relationship with the Solomon Islands.
"The delegation fully accepts this point."
The MSG delegation originally planned a media conference at the PNG High Commission in Canberra today, but later called off the event - instead issuing a statement saying it held "constructive and positive dialogue" with the Australian officials.
They said their mission to both countries was not as mediators.
The MSG delegation included PNG's foreign affairs secretary Gabriel Pepson, Fiji's foreign affairs chief Isikeli Mataitoga, and the head of Vanuatu's prime minister's department, Jean Sese.