The East Dunbartonshire MP raised concerns over the "rise of nationalism" in Scotland and across Europe, but insisted her party can become as "rallying point" for liberal values which voters are turning to in response to Brexit and extremist political movements.
The Lib Dems swept back to the centre of British politics with widespread gains at the European elections last week across the UK on an anti-Brexit platform attracting moderates from Labour and the Tories.
But the SNP victory in Scotland saw Nicola Sturgeon publish fresh legislation paving the way for a second independence referendum.
"It's really disappointing and sadly not surprising that the SNP are back banging their independence drum," Ms Swinson told the Scotsman.
"Barely had the ballot papers been counted before she's back again with her independence obsession and trying to use that result as a suggestion that Scotland wants to have another independence referendum when the polls show that that's not the case and Scotland comprehensively rejected a specific plan for independence in the referendum in 2014.
"That's why we want Scotland to be part of the family of nations of the United Kingdom and why we're fighting so hard for the United Kingdom's place in the European Union."
The European elections saw parties campaigning for Remain secure 40% of the vote UK-wide, compared to the hard pro-Brexit parties getting 34% across the UK.
"This fight is not over," Ms Swinson added.
"The SNP shouldn't be packing up things and running away at this point when the battle stop Brexit happening absolutely is still ongoing.
"They're saying that they don't think the union with the other nations of the United Kingdom is one that is positive and I wholeheartedly disagree.
"Whether we're looking at our economy, whether we're looking at 300 years of shared heritage and history, whether we're looking at the links with families that we have across the border, we're part of one United Kingdom and that is something which is to be treasured and it looks like again to fought for."
Ms Swinson lost her seat to the SNP's John Nicolson in 2015 before recapturing it two years later. She is married to former Lib Dem MP Duncan Hames and the couple have two sons.
And despite the prospect of a hardline Brexiteer leader of the Tory party taking over from Theresa May, Swinson believes that a the strength of anti-Brexit feeling in the Euro elections means a second Brexit referendum is within reach and could be staged as early as next year.
"More people will really feel the need to put their country ahead of their party's interest," she added.
"We've seen an unprecedented number of resignations from this Government many of them over the issue of Brexit and I think as minds focus on the decisions in the forthcoming months, then there is every chance that we can secure that People's Vote.
"If the SNP have given up on that, I think that's a huge shame. I think is letting down people who in Scotland want to see us stay in the UK and the UK in the EU."
The 39-year-old former business minister in the 2010-2015 Coalition Government is the favourite to succeed Vince Cable as party leader, but will face competition from Ed Davey, the party's former energy secretary.
Swinson says the party must win "a lot more" than the 11 seats in the last election as it was made to pay for its role in the austerity politics of the Coalition with the Tories.
She added: "We are on the up and in this time when you see the rise of nationalism in lots of parts of the world, including in our country, there is a need for a liberal voice, a liberal alternative, like never before.
"It's very clear that the Liberal Democrats are the rallying point for those liberal values and we continue to grow that liberal movement right across the country."
"The current times that we're operating in provide us with more opportunity to grow faster than that and also a greater imperative that we do. You're looking at a politics where you've got on the one hand hard-left Labour leader who doesn't have solution fit for the 21st century and on the other hands you've got a Conservative party whoever they choose, are going in a more right-wing direction, listening to the siren calls of Nigel Farage and his nationalism and that agenda."
She added; "The opportunity is with those big parties veering off to those extremes there are so many people who are crying out for a positive liberal alternative and we need to fill that space.
"I believe as leader that Liberal Democrats will be a rallying point for a much bigger movement."