May took the reins of government on Wednesday, replacing David Cameron, who quit in the wake of the referendum.
Scotland needs the stability and security of remaining in the world's biggest single market - for jobs, investment and prosperity.
"You can not uphold the European Union referendum result in Scotland as sacrosanct while trashing the Independence referendum vote we held just 22 months ago". While she campaigned for "Remain", she has said repeatedly that "Brexit means Brexit" and followed up by appointing prominent "Leave" campaigners to head the departments that will be most involved in the Brexit talks.
He said: "I think the best future for Scotland is inside the United Kingdom economy".
But she has been slammed by Scots Tory leader Ruth Davidson, who was at David Cameron's final Cabinet meeting this morning to say goodbye to him but also to talk to the incoming Prime Minister about taking Scotland and the United Kingdom forward.
May's decision to visit Sturgeon on her own turf less than 48 hours after taking office underlines her determination to keep Scotland in the United Kingdom after the Brexit vote revived the issue of independence.
Commenting, SNP MSP Joan McAlpine, who is Convener of the Scottish Parliament's European and External Relations Committee, said: "These are very encouraging figures showing that people across the continent value Scotland's role in Europe - and from France and Germany to Finland and Sweden, it's clear that there is strong support right across Europe for Scotland's EU membership".
After six years as Cameron's interior minister, May was viewed as a safe pair of hands to replace him, but began with a deep cull of some of her former cabinet colleagues. As a high level cabinet Minister (May was the longest serving Home Secretary for over a century), this is not surprising. "Whether it's reforming the economy or strengthening our society, we are going to build a better Britain and a nation that works for everyone; not just the privileged few", she added.
"March of the meritocrats", was the headline on the cover of the Daily Mail tabloid.
Meantime, the PM, who has already taken telephone calls from Germany's Angela Merkel and France's Francois Hollande, had a 15-minute congratulatory telephone call from US President Barack Obama.
Theresa May is using her first official visit as prime minister to try to persuade Scotland that there is nothing to fear in a post-Brexit U.K.
The Bank of England on Thursday held off on an anticipated cut in interest rates, keeping them unchanged at 0.50 percent, but signalled a possible cut next month.
She highlighted the problems that the ECHR had made for her as Home Secretary, including delaying the extradition of Abu Hamza, almost stopping the deportation of Abu Qatada, and when the ECHR "tried to tell Parliament that - however we voted - we could not deprive prisoners of the vote".
Meanwhile, the funeral of murdered Labour MP Jo Cox was to take place in her northern Yorkshire constituency later Friday.