Mrs May appointed junior minister Dominic Raab, a former diplomat who was a prominent campaigner for Leave in the 2016 referendum, as Mr Davis's successor.
He added that Boris Johnson was "a great friend to Australia" and Foreign Minister Julie Bishop in particular had a good relationship with him.
He told BBC Radio 4's Today: "It seems to me we are giving too much away too easily and that's a unsafe strategy at this time". "And that's a risky strategy at this time". Mr Davis's deputy, Steve Baker, also resigned.
The two resignations have left May badly exposed and raised questions over whether she will stand firm in her commitment to pursuing a "business-friendly" Brexit, or will be faced with more resignations and calls to quit herself.
"Being tied to European Union regulations and the European Union tying our hands when seeking to make new trade agreements will be the worst of all worlds", wrote the Mansfield MP, who voted remain in a constituency where more than 70% of voters opted to leave.
May's proposals for a future European Union relationship after Britain departs from the bloc next March had taken two years of internal government wrangling to agree.
"But, I think it's reasonable to set out the very worst case scenario for a five to 10-year period to iron out all the details of a trade deal".
Defence Secretary and South Staffordshire MP Gavin Williamson said: "The Prime Minister is putting her best foot forward in making sure we get the best deal with the EU".
"Over the coming days, weeks and months these negotiations will become more intense as we head towards leaving the European Union on March 29, 2019".
They are planning a show of strength in the Commons next week in votes on the Prime Minister's flagship Trade Bill.
The resignations came just days after May announced Saturday that she had finally united her quarrelsome government behind a plan for a divorce deal with the EU.
The Eagles' hit song ends with the ominous lyrics: "You can checkout any time you like, but you can never leave".
As many as 60 Tory rebels could back moves opposing key elements of Mrs May's Brexit strategy - more than enough to wipe out her fragile parliamentary majority.
All of this is huge trouble, and bookmakers are all circling like hawks to offer political betting fans yet more specials, with a stacked portfolio of Tory betting specials already up.
Asked if the prime minister could survive.
Since then, two vice chairmen of the party have also resigned (Ben Bradwell and Maria Caulfield) have also resigned stating that they can not support the current government position and fear that their constituents would vote them out of parliament should the nation be faced with a general election on the back of the current Brexit position.