On Wednesday, the outgoing Welsh Conservative leader said there was "plotting" taking place as long ago as April.
That was denied by Peter Fox, one of the seven MSs that voted against him.
"There was a range of things that happened over recent months that many of us were starting to feel uncomfortable with," he said.
Millar said he has wants to build on Davies' legacy and "get back to holding the Welsh government to account - that has to be our focus".
Under rules that require three other MSs to back a candidate, unless someone switched support, a rival candidacy is impossible.
Grassroots Newport Tory Huw Davies said it was "very disappointing" that Conservative members will not be able to elect the new group leader.
"There are a number of issues which need raising, and it will leave the new leader without a mandate," he said.
Nominations close at 17:00 GMT on Thursday.
A Welsh Liberal Democrat spokesperson said: "In July Welsh voters delivered their devastating verdict on the Conservatives, booting out every single Tory MP in Wales.
"Re-arranging the deckchairs in their Senedd group isn't going to make anyone forget their record of incompetence, sleaze, and failure."