He said: "There was no outright victor, the SNP have one more member, and that means we have to have a totally different mind-set".
Labour's former council leader Frank McAveety, who retained his seat in Shettleston, said: "The SNP predicted they could return 56 councillors, they've only returned 39 councillors".
Labour has already lost overall control of Glasgow City Council after the first three results were declared.
Nicola Sturgeon hailed the Scottish local elections as an "emphatic" win for the SNP - despite the Tories gaining 167 seats and returning a record number of councillors.
Ms Davidson has described Moray as "very pro-Union" and emphasised the fact it is one of the few pro-Brexit parts of Scotland, placing many constituents at odds with Mr Robertson who she accused of "hi-jacking" the EU referendum result as proxy support for independence. "We know from the polls in England that Theresa May is going to win this election, the question is who is going to stand up and make Scotland's voice heard?"
Last week's results followed a campaign by the Tories focused on opposition to a second referendum - a message they will carry through their General Election campaign in the weeks ahead.
COSLA rules require that the top two lay posts are both gender-balanced and occupied by different parties, with the result that the Tories, having pushed Labour into third place across the country, may insist on one of the posts.More news: At Least 25 Killed After Blast Hits Convoy Of Senior Pakistani Official
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"So they put that issue centre stage and they lost the election". Although that marked the first time that the SNP had managed to outpoll Labour in local elections, the outcome was, in truth, widely regarded as a disappointment for the nationalists.
Meanwhile in East Garioch returning councillors Martin Ford (Scottish Green Party) and Fergus Hood (Scottish Liberal Democrats) will be joined by Dominic Lonchay (Scottish Conservative and Unionist) and Glen Reid (SNP).
"They came second in the election and the SNP came first".
She is expected to say: "Theresa May's extraordinary attacks on our European partners demonstrate that an unchecked Tory government is prepared to pursue a chaotic hard Brexit if it is in the interests of the Tory Party, whatever the cost to Scotland".
Campaign for Socialism spokesman Martyn Cook said: "Labour is being squeezed between two increasingly assertive nationalisms, neither of which is able to deliver for working people". It just has to stay on the pitch, hold its ground and people will come back to us.
And this, I suggest, is a problem for the first minister. Fisheries will go, there will be movement on immigration, on freedom of movement - she will buckle on that.
"They've had a good day by their standards but we've got to put it into context. Ukip could be bigger than it ever was before".