The newly-elected party leader defended her record in the coalition but rejected the possibility of supporting a government under Conservative leadership frontrunner Mr Johnson.
Ms Swinson said that she was "crystal clear" that the Lib Dems could not work with any party who supports Brexit and indicated she would prefer a people's vote on EU membership to a general election.
The Lib Dems' first female leader said that "the rules of politics are being rewritten" and added that she hoped to get a majority in the House of Commons for a people's vote with the support of Tory ministers and MPs who have ruled out serving under Mr Johnson in opposition to his Brexit stance.
The successor to Sir Vince Cable also said that - if a people's vote did take place and a leave option won - she would "absolutely respect" the result, arguing that no-one voted for a "specific Brexit deal" in the first EU referendum.
Asked on BBC Scotland about her backing for a second European referendum but not another vote on Scottish independence, the 39-year-old claimed that the no vote in 2014 was against a detailed proposal "that was comprehensively rejected", whereas a people's vote was a "way out of this current mess and crisis".
"Chaos in our politics is stopping us from moving on and addressing the real issues our country faces, so there is actually a way out of chaos," the MP for East Dunbartonshire said.
"If you were to have an independence referendum in Scotland, that is layering chaos on top of that already volatile and difficult situation."
Confirming that she would say no to SNP demands for another referendum were she to become prime minister, she added: "There is no mandate to have that. The SNP have put forward this idea of IndyRef2 at subsequent elections and they have lost seats and lost votes.
"In Scotland, most people want Scotland to be in the UK and the UK to be in the EU, and the Liberal Democrats are the only party that are arguing for that position."