John Swinney said the new Prime Minister's determination to take the country out of the EU at all costs by October 31 proved how little influence the Scottish Tory leader now had at Westminster.
The Holyrood education secretary made the comments while visiting Shetland yesterday ahead of a Scottish Parliament by-election on the islands next month.
Mr Johnson, meanwhile, was touring the Faslane naval base on Monday before heading to Edinburgh for talks with Ms Davidson at Holyrood, where the pair are said to have brokered an uneasy truce.
Davidson has publicly declared she could not back the new PM if he goes through with his threat to carry out a No Deal Brexit. She backed three of Johnson's rivals in the race to become Tory leader after Theresa May stepped down last month.
“I think Ruth Davidson is in an absolutely impossible position," Mr Swinney told the Shetland News. "She has said that she does not support a no deal Brexit, but her Prime Minister is careering towards one.
"I think it illustrates first of all the lack of influence that Ruth Davidson has on the UK government now, and secondly, it demonstrates the need for Scotland to make sure that our interests are protected through this in all our political interests, and it makes it absolutely abundantly clear that our political interests will not be protected by voting for the Conservative party.”
The Scottish Tory leader, speaking after her meeting with Mr Johnson, said the prime minister had made clear the UK Government's preference was to leave the EU with a deal.
But Swinney said it was clear the new administration at Downing Street was preparing to crash out with no deal in place.
He continued: "That’s what he’s planning for, that was made very clear over the weekend, and I cannot understand how any rational government acting supposedly on behalf of the best interests of the public could come to the conclusion that a no deal Brexit is what we should have inflicted upon us.
"This would do incalculable damage to our economy. So I think what Boris Johnson needs to hear from the people of Scotland is that as a country we voted overwhelmingly to remain, every single local authority area in Scotland voted to remain, in the European elections the SNP won in every local authority apart from the Northern Isles, where we were very closely behind the liberals, who again support remain. So Scotland could not have been clearer in our desire to remain within the European Union, and the Prime Minister needs to take account of that in all the decisions that he takes.”
A Number 10 spokesman said: "The Government’s preference is to negotiate a new deal which abolishes the anti-democratic backstop, the UK will be leaving the EU on 31 October come what may."