Opposition leader Jeremy Corbyn nonetheless showed no sympathy for the outgoing prime minister, who is expected to step aside on 7 June. She cannot govern, and nor can her divided and disintegrating party.
Whoever succeeds May as Conservative Party leader will nearly certainly have to demand a tougher Brexit deal from Brussels - yet the European Union has repeatedly said it will not rework the Withdrawal Treaty.
"Parliamentary procedure offers no route, and the only apparent way to blocking "no deal" - a vote of no confidence - would be a massive gamble for (Conservative members of parliament)", wrote the non-aligned Institute of Government.
The First Minister said: "I wish Theresa May well".
"As leader of the Scottish Conservatives, I want to see candidates show that same level of commitment to Scotland's place in the union, an ability to advance our interests at home and overseas and, crucially, demonstrate how they intend to bring our country back together after the divisions sown by two constitutional referenda".
"The second female prime minister, but certainly not the last".
Reacting to Mrs May's resignation statement, Ms Sturgeon said the the prospect of a "hardline Brexiteer" becoming the next Prime Minister was "not a happy one" for Scots.More news: Tornadoes kill three, hit U.S. Plains state capital
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The Prime Minister had hoped her deal, which opens up the possibility of a second referendum, would sway Labour to get behind it.
The end of Theresa May's premiership will usher in an even more turbulent phase of Britain's exit from the European Union as any new leader is likely to seek to strike a tougher divorce deal, and there could be an election within months.
"Her departure will not solve the Brexit mess that the Tories have created", Ms Sturgeon said.
She added: "Only putting the matter back to the people can do that".
This will trigger a leadership contest in the party with the victor becoming the next British prime minister. "Given current circumstances, it also feels deeply wrong for another Tory to be installed in Number 10 without a General Election". "Not only has that been completely ignored, but it is now clear that a devastating no-deal Brexit - which would cost jobs, businesses and livelihoods - is firmly back on the Tories' agenda", she added.
International Trade Secretary Liam Fox tweeted that Mrs May "acted with dignity and honour in pursuit of what she believes to be in the national interest, the hallmark of her time in public life".
"Added to the experience of the past three years, this makes it all the more important that Scotland is given the choice of becoming an independent country".
"I do so with no ill will, but with enormous and enduring gratitude to have had the opportunity to serve the country I love".