The Scottish Conservative leader said the local authority results were a message to both the Tories and Labour after they failed to reach a deal on leaving the EU.
She spoke out on the issue as she launched her party’s campaign for the ballot at Deanston Distillery near Stirling today.
Ms Davidson said: “I think it’s clear that people are frustrated that Brexit hasn’t happened yet, because they were told it would.
“The council elections down south were a plague on both your houses – both Labour and the Conservatives – for not reaching a deal.
“I would encourage both my colleagues in the Conservative party in the House of Commons and the Labour Party to try to walk back to the centre and let’s get a deal over the line.
“The people of this country are watching and they’re giving them a pretty clear instruction.”
Ms Davidson said voters could trust her party not to re-run any referendums.
She urged MPs to give serious consideration to Theresa May’s deal when it is brought back to Parliament in the first week of June.
Ms Davidson said: “I would urge colleagues from all parties in the House of Commons to consider very seriously the deal that’s brought forward next month, when the Prime Minister brings it back.
“Yes, we’ve heard all of the things that people don’t want, but we do have to coalesce around something that the country does want, that it can live with, that will give us that orderly Brexit, that will allow companies like the one I’m visiting today to have answers and to make decisions about their future.
“The alternative is the EU27 doesn’t grant any further extensions, that we fall out without a deal.
“That would be incredibly disruptive to some of the businesses that I’ve been speaking to across Scotland.”