Ian Blackford said any proposition to halt Brexit by revoking Article 50 would have to have teeth and currently it is not clear that would be "particularly easy" to do.
He said a no-deal Brexit would "crash the economy" and that Theresa May's resignation means an increased threat of leaving the EU without a deal as a default on October 31.
"I think, at the end of the day, there seems to be a very real risk to the people of Scotland, not just the people of Scotland but the people of the United Kingdom, in terms of that no-deal scenario."
He called for a general election, a people's vote and for Article 50 to be revoked.
"What Theresa May was talking about was bringing forward a Bill that would be become an Act of Parliament and of course that could be amended and then would give us the option to revoke Article 50 but we simply don't know at this stage what a new prime minister will do."
Asked if any prime minister determined to take the UK out of the EU could do so by the default date by not doing much at all, Mr Blackford said: "I'm afraid that is the case. I know that sounds absolutely ridiculous but ... that is the default position.
"That's where we are from the legislation that is already in place so we have to find a way to stop that.
"The best way of doing it, in many respects, is if the new prime minister does bring forward legislation, but on a no-deal basis that it's possible they could avoid that.
"So we are in a very dangerous, very worrying situation in terms of our future as members of the European Union."
Speaking on the same programme, Conservative MP Douglas Ross said all candidates for Tory leadership support a Brexit deal and accused the SNP of being "hypocritical".
He said: "They (the SNP) have had opportunities to leave the European Union with a deal and voted against that on every occasion."
Questioned on the ability to negotiate an amended Brexit deal with the EU, Mr Ross said he believed with a new prime minister following the European election could "see some progress".