Mr Robertson formally launched his bid to become depute leader of the SNP with a warning to Mrs May that her days as leader of the whole of the United Kingdom will be numbered if she fails to respect Scotland’s desire to remain part of the European Union.
Almost two thirds (62%) of voters north of the border backed keeping the UK in the EU in last month’s referendum, with SNP leader and First Minister Nicola Sturgeon saying afterwards that the result makes another vote on Scottish independence “highly likely”.
Mr Robertson, the SNP leader at Westminster, hit out at Mrs May after she said on a visit to Edinburgh that Scots have “had their vote” on independence in 2014.
In a speech, Mr Robertson told an audience in the Scottish capital on Thursday evening: “Just last week, Theresa May said that options for keeping Scotland in the EU were ‘impractical’, and that we’ve ‘had our referendum’.
“I think we have an opportunity to show her that things are different in Scotland.
“She has of course said that there will only be a ‘UK approach’ to Brexit. She has refused to accept that for Scotland, Remain means Remain.
“My message to the Prime Minister is this: If you ignore the expressed will of the people of Scotland, if you refuse to even consider how we might protect Scotland’s place in the EU then be in no doubt - your days as Prime Minister of a United Kingdom are numbered.”
Mr Robertson added that the next few months will be a “major test” for the SNP - which now has more than 120,000 members - with the party facing the prospect of “delivering a new prospectus for independence”.
The Moray MP said: “We are truly on the brink of independence - the campaign that comes now should be all about persuading people why our vision for Scotland is the right one.”
Mr Robertson said that the Brexit vote has brought “much uncertainty” with it, but said Scotland and the party have a future “redolent with opportunity”.
He told SNP members and activists: “Our next job is to communicate with the many people who believe their 2014 referendum vote was to stay in Europe by voting ‘no’ to Scottish Independence.
“They are right to feel cheated. They are right to feel betrayed.
“After that, we need to speak to those many people who thought they were voting for certainty over risk.
“Since 2014 it’s the UK that has become the risky option, and since the Brexit vote this is about to get much much worse.
“We need to take our new prospectus to the streets to make sure that we can deliver the best for Scotland.”
Mr Robertson outlined three themes to his campaign: leadership, engaging grassroots members across Scotland, and independence.
The MP, who joined the SNP as a teenager in the 1980s and was part of the team which helped it become an “electable victorious party of government” in 2007, said he has the necessary skills and track record to be depute leader.
“I’ve learned a lot about leadership over the last year, leading a talented group of MPs, the largest ever team of SNP MPs in Westminster,” he said.
“We’ve gone from six MPs to the third largest party in Westminster. I am so proud of our team, they have achieved so much.”
And in a swipe at the official opposition, he added: “We’ve provided the leadership whilst Labour has had none.”
Mr Robertson’s most high-profile challenger for the depute leadership is Alyn Smith, the MEP who received a standing ovation in Brussels for his plea to EU members to respect Scotland’s vote to remain.
Mr Smith has offered to be ‘’a freelance roving sherpa’’ for Ms Sturgeon in Europe to get the best deal for Scotland.
Other contenders are MP Tommy Sheppard and councillor Christopher McEleny.