Ms Davidson, tipped by many as a future UK Tory leader, hailed a "historic" night for her party and said the result ended the prospect of another independence vote soon.
"The SNP tsunami of 2015, in which the party took 56 of Scotland's 59 constituencies, was followed yesterday by an natural disaster in which the SNP lost more than a third of their MPs", says Alex Massie in The Times.
Ms Sturgeon said "Undoubtedly the issue of an independence referendum was a factor in this election result, but I think there were other factors in this election result as well". Both parties embraced "valence politics" wholeheartedly, focusing on each other's lack of governing competence (the SNP in Scotland, the Conservatives in the UK), accentuating focus on the leaders, and simplifying their message to the extent that if there was any discussion of actual policies going on, they had failed.
"They recklessly force through an European Union referendum, they then embarked on a disastrous Brexit strategy, deciding to remove Scotland and the United Kingdom from the single market with no idea and no plan for what would come next".
The reports were seen as a sign that the party's Scottish leader was seeking to assert the influence of her MPs.
In a general election full of unexpected twists, perhaps the biggest came north of the border where the Scottish National Party suffered heavy defeats ending their one-party hegemony. In many instances this was the Conservative candidate, as the party has successfully positioned itself as the most strident defender of the Union under the leadership of Ruth Davidson.
"Once we know what the final result is tomorrow ... we will want to, if we possibly can, try to be part a progressive alliance which is an alternative to the Tories". Now she can say "if I pull my 13 away you don't have a majority". I'm afraid that's not enough.
Sturgeon demanded a second independence ballot in March, arguing that the Brexit result changed the rules of the game.
Ms Davidson said she had received assurances from Mrs May that there would be no "rowing back" on LGBTI rights in return for DUP votes in the Commons.
Reflecting on the overall British results, Davidson said her Conservative Party needed to take account of their losses: "It is incumbent on us to listen to other parties in Parliament, and people outside it, about the best way forward".
Ms Davidson said the UK Government must "seek to deliver an open Brexit, not a closed one, which puts our country's economic growth first".