Sturgeon is pressing for a referendum in the second half of 2018 or the first half of 2019, before Britain leaves the European Union, but May has told her "now is not the time".
The Edinburgh assembly's vote, which was widely expected, gives Sturgeon a mandate to seek permission from the British parliament in London to press forward with preparations for a referendum.
Ms Sturgeon told Sky: 'She's very clear that she thinks the terms of Brexit, both the exit terms and the details of the future relationship - the future trading agreement between the United Kingdom and the European Union - the detail of that will be clear before the United Kingdom exits.
Following several hours of debate, MSPs voted by 69 to 59 to back the SNP's demands for a so-called "Section 30" order allowing the Scottish government to stage another poll.
The former deputy prime minister said there was a "very deliberate attempt" under way by hard Brexiteers to use issues like the £50 billion "divorce bill" expected from Brussels as excuses to ditch these promises and drive the United Kingdom out of the European Union without a deal, with the intention of transforming Britain into a low-regulation, low-tax haven off the shore of Europe.
"While negotiations continue we will continue to be the "guard dogs of Brexit", holding the Government's feet to the fire at home and our MEPs will be working to ensure our friends on the continent do not pull any fast ones. Pulling together to make sure we get the best possible deal for the whole of the U.K.", May said Monday.
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Scottish Conservative leader Ruth Davidson said the decision to go ahead with another vote was "deeply regrettable", while Scottish Labour leader Kezia Dugdale said there was no evidence that Scots want another referendum.
The Scotsman newspaper in Scotland reported that when asked about what would happen if Sturgeon's call for another referendum was formally rejected, the SNP leader replied: "I will set that out in due course".
But Sturgeon says Brexit has changed the situation dramatically.
She said: "There absolutely should not be another independence referendum until after Brexit".
The prime minister added: "My message is clear - now is not the time".
Labour, the Conservatives and the Liberal Democrats said Sturgeon was neglecting the parlous state of Scottish public services in her pursuit of an unpopular constitutional obsession.
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The debate over the second referendum began last week but was suspended on 22 March after the terrorist attack in Westminster.
But the referendum has the backing of the Scottish Green Party, which holds six seats.
Britain as a whole voted to leave the bloc in a referendum previous year, but Scots voted by a large margin to stay.
The most important and basic principle is that the fundamental powers of a sovereign independent country will be returned to the British people as represented in our democratically elected parliament. If it does so, I will enter discussion in good faith and with a willingness to compromise. "If their (May's government) position remains as it is right now, I will set out to parliament what I think the next steps should be".
I agree with that, which is why I have indicated a timescale no earlier than 18 months from now, when the terms of Brexit are clear - something the PM has now indicated she agrees with.
Scotland voted against independence in 2014.
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