In a speech at the Scottish Tory conference in Glasgow, the Prime Minister said "politics is not a game" and also condemned the Scottish National Party's record in government on education.
"A tunnel vision nationalism, which focuses only on independence at any cost, sells Scotland short," said Mrs May.
"People in Scotland deserve a First Minister who is focused on their priorities - raising standards in education, taking care of the health service, reforming criminal justice, helping the economy prosper, improving people's lives.
"Instead, they have an SNP Government obsessed with its own priority of independence, using the mechanisms of devolved government to further its political aims and all the while neglecting and mismanaging public services in Scotland."
But the First Minister has already hit back, claiming her attempts to reach a Brexit compromise have been met by a "brick wall of Tory intransigence".
The latest clash between Mrs May and Ms Sturgeon comes as the First Minister is expected to announce plans for a second independence referendum when the PM triggers Article 50 for the UK to leave the EU later this month.
Mrs May claims it is "very clear" that people in Scotland do not want another vote on independence, but has so far not said whether she will grant permission for it if the Scottish Parliament calls for another referendum.
In her speech, she issued a strong defence of the Union and urged her party to "never stop making loudly and clearly, the positive, optimistic and passionate case for our precious union of nations and people".
"We are four nations, but at heart we are one people," she said.
"That solidarity is the essence of our United Kingdom and is the surest safeguard of its future."
She said the UK domestic market was worth four times more to Scottish firms, with the EU coming in third after the rest of the UK and the rest of the world as a market for Scottish goods.
"And yet the SNP propose Scottish independence, which would wrench Scotland out of its biggest market," she said.
Mrs May said was determined to achieve a Brexit deal which "works for all parts of the UK - England, Scotland, Wales and Northern Ireland - and for the United Kingdom as a whole".
She said: "I am confident about the future of our United Kingdom and optimistic about what we can achieve together as a country.
"There is no economic case for breaking up the United Kingdom, or of loosening the ties which bind us together."
And in an attack on public services, Mrs May said Scotland's schools are "outperformed in every category" by those in England, Northern Ireland, Estonia and Poland.
"As Prime Minister of the United Kingdom, I am just as concerned that young people in Dundee get a good start in life and receive the education they need to reach their full potential as I am about young people in Doncaster and Dartford," she said.
"The SNP's neglect and mismanagement of Scottish education has been a scandal.".
Alex Salmond slammed the address as "arrogant and high-handed" and accused Mrs May's government of being "consumed with Brexit".
He said while Scotland had its full share of challenges in public services, they were "performing much better than" those in England.
He told Sky News: "55% of Scottish kids go on to higher education - in England the figure is 45%
"Let's remember, the English health service is in a state of collapse, the prison service in England is in total chaos, the schools are revolting while Theresa May's government is consumed by Brexit.
"The verdict on Nicola Sturgeon is the verdict of the Scottish people, not the judgement of the unelected Theresa May."
He said if Mrs May were to reject Mrs Sturgeon's attempts to reach a Brexit compromise then a second Scottish independence referendum "becomes odds-on", adding: "And the odds are shortening by the day".
"The days of Scotland being lectured to by high-handed prime ministers at Westminster, these days are over.
"Theresa May better become accustomed to that new reality very quickly or she'll be facing an independence referendum in Scotland," he added.
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