It means the MSP for Edinburgh Eastern becomes the Alba Party’s first representative in the Scottish parliament, though it has two MPs at Westminster.
The SNP called on Ms Regan – who lost out to Mr Yousaf in the battle to succeed Nicola Sturgeon in March – to stand down so a by-election could be held.
In a statement, Ms Regan said: “I could not, in good conscience, continue to be part of a party that has drifted from its path and its commitment to achieving independence as a matter of urgency.”
“Today, I am proud to take up the mantle of leadership for Alba at Holyrood and to become the first Alba MSP,” she added.
Ms Regan said she wanted to provide “a clear focus on reinvigorating the cause of independence and delivering on the promise that was made to the Scottish electorate in 2016 and 2021”.
It follows a series of woes for Mr Yousaf, with the latest political hit coming from the defection of MP Lisa Cameron to Rishi Sunak’s Conservatives at Westminster.
The SNP accused Ms Cameron of “betrayal” for switching to the Tories – questioning whether she ever believed in independence and demanding that she “do the honourable thing” and resign her seat.
Mr Salmond, the former first minister and SNP leader, started his own breakaway party after falling out with Ms Sturgeon, said he was “delighted” to welcome Ms Regan to “team independence”.
Speaking at the start of his party’s conference, he added: “Her commitment to the cause of Scottish independence has never been in question, and her addition to Alba sends a powerful message about the focus and determination we bring to achieving an independent Scotland.”
Alba, who not yet won an electoral seat, has adopted a more strident approach to Scottish independence than the SNP.
Mr Salmond says he would demand negotiations with the UK government on independence, rather than seek another referendum, if there was majority for pro-indy parties at the next election.
An SNP spokesperson said: “Ash Regan should do the honourable thing and step down to allow a by-election.”
They added: “Her constituents elected an SNP MSP, and they deserve to have the democratic opportunity to elect a new, hard-working SNP parliamentarian who will put the interests of Scotland first.”