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* Cameron to offer more tax and spending powers
* Scottish nationalist leader wants to go further
* Nationalists won 56 of 59 Scottish seats in UK parliament
By Paul Sandle, Alistair Smout and Andy Bruce
EDINBURGH/ABERDEEN, Scotland, May 14 (Reuters) - Prime Minister David Cameron travels to Scotland on Friday to face his new adversary, the Scottish secessionist leader whose triumph at last week's election put Britain's unity in jeopardy just nine months after a referendum saved it.
As both sides brace for a confrontation, Cameron is offering Scotland the "the strongest devolved government anywhere in the world", with more powers to set income tax rates and influence welfare spending.
That is unlikely to satisfy Scottish National Party leader Nicola Sturgeon who wants complete control over tax and spending, and potentially another chance to win the independence Scottish voters rejected last year.
"We want to see devolution of powers over employment policy, including the minimum wage, welfare, business taxes, national insurance and equality policy," Sturgeon told the Scottish Parliament in Edinburgh.
She said she wanted Cameron to look again at London's proposals, which were drafted with the agreement of all Britain's traditional parties, with "a with a view to extending devolution even further."
Last week's general election saw Sturgeon's secessionist party win 56 of the 59 Scottish seats in Britain's House of Commons. Cameron's Conservatives won just one.
The SNP campaigned from the left and argues that its victory shows Scottish voters have decisively rejected the public spending cuts imposed by Cameron's Conservatives. Sturgeon argues that any future settlement must give Scotland the power to pursue the less austere policies Scottish voters demand.
Cameron, a distant cousin of Queen Elizabeth, is as clear that he doesn't want to preside over the collapse of the United Kingdom as Sturgeon, daughter of a Scottish electrician, is clear that she still wants independence one day.
They have already publicly clashed over a new referendum: Cameron says there won't be one. Sturgeon says that is not Cameron's call but a decision for Scots.
Her pugnacious predecessor, Alex Salmond, 60, who stepped down as SNP leader after last year's referendum and will now take up a seat as a newly elected SNP lawmaker in London, told reporters he still expects to see independence in his lifetime.
"This is the latest staging post in what seems like progress for the Scottish people."
Scotland's three centuries old union with England could depend on the fate of another union: Cameron has promised to renegotiate Britain's relationship with the European Union and hold an in-or-out EU membership referendum by the end of 2017.
Sturgeon has said that if England voted to leave the EU but Scotland voted to stay, it could trigger another referendum on Scottish independence.
For now though, a new chance at independence is not yet on the cards. The focus instead is on finding a role for Scotland within the United Kingdom.
After the independence vote, Britain's main parties agreed to grant Scotland more tax-raising powers as part of the biggest transfer of authority to Scotland from the United Kingdom since 1999 when a Scots parliament was set up.
The SNP holds the most seats in the Scottish parliament and Sturgeon leads Scotland's government as first minister. Her government already controls most state services such as education and health, but has little say over how much Scottish taxpayers pay and it gets to spend.
Under draft laws which Cameron is expected to pass, Scotland will have control over tax revenues worth 20 billion pounds a year and welfare spending worth 2.5 billion pounds a year, although some taxes the SNP wants control of are excluded.
"The government has already said there will be very significant further devolution, there was cross party agreement on that," a spokesman for Cameron said.
"The prime minister of course will regularly meet and discuss with the first minister of Scotland. I think it is fair to characterise things as getting on with the further devolution that has been promised to the people of Scotland."
Cameron's unexpected majority in the UK parliament limits Sturgeon's influence over UK policy. Instead of wielding power as kingmakers in London's Westminster parliament as polls predicted ahead of the vote, her cadre of lawmakers will sit on the opposition benches with the defeated Labour party.
Cameron's aides say it is now up to the SNP to decide whether to play a part in helping to shape the new settlement offered to Scotland, or sulk in opposition.
"They are the third largest party in the House of Commons: The ball's in their court whether they use that constructively or destructively," said a government adviser, speaking about party politics on condition of anonymity.
The SNP is due to face Scottish voters once again in an election for the Scottish parliament next year. Some voters say Sturgeon's new mandate means she has no excuse if she fails to deliver on her promises.
"I voted 'No' in the referendum, but now they [the SNP] have five years in which they can prove themselves and they can deliver without blaming everything on Westminster," said Eilidh Maclean, a 19-year-old IT worker in oil city Aberdeen.
"This is Nicola Sturgeon's time, she can't blame anyone else, she's got all the power that she needs to deliver everything that she promised. And if she can't do that, she can't deliver on independence." (Additional reporting by Kylie MacLellan in London; Editing by Peter Graff)