Britain warned on Friday that leaving the European Union without a divorce deal would mean drugmakers needing to stockpile experimental treatments from remaining member states in case of border delays.
Theresa May will be given one final chance to sell her Brexit plan to the European Union on Wednesday - with a warning from Brussels that no deal is "more likely than ever before".
Writing in the Sunday Times, Mr Davis, who quit in July over Brexit, said: "It is time for the Cabinet to exert their collective authority".
This weekend The Sunday Times reported nine cabinet ministers were dissatisfied with Theresa May's Brexit stance and want her to change course.
Other allies in the Democratic Unionist Party and opposition MPs in Labour and the Liberal Democrats also have doubts, which will make it very hard for Mrs May to get parliamentary backing for any Brexit deal.
May has insisted that she would never agree to the EU's proposal, which would separate Northern Ireland constitutionally and economically from the rest of the UK.
"We can not let this disagreement derail the prospects of a good deal and leave us with a "no-deal" outcome that no-one wants", she told parliament.
But negotiations ended without a breakthrough, including on the issue of trade to and from Northern Ireland, which has emerged as a possible deal breaker and even a threat to Prime Minister Theresa May's leadership.
So, the backstop is a position of last resort - to protect an open border on the island of Ireland in the event that the United Kingdom leaves the European Union without having agreed a solution as part of trade negotiations.
But lobbying of May from all sides has increased in recent weeks as London and Brussels edge closer to an agreement on a draft withdrawal treaty to cover the divorce terms, a transition period and a solution for Northern Ireland.
Then on Wednesday, May goes to Brussels, still facing the issue of the border between Northern Ireland and the Irish Republic.
Johnson, the bookmakers' favourite to replace May, is the latest critic to redouble efforts to urge the prime minister to rethink her plan to leave the EU.
He said Brussels "continues to insist" on the possibility of a customs border down the Irish Sea.
But a spokesperson for May said there were "a number of means of achieving what we want to achieve" on the backstop.
The PM was thought to be trying to break the deadlock by proposing a new "backstop" arrangement for the Irish border.
"The decisive factor was enormous pressure from her own Cabinet, which compounded the problems she already had with Eurosceptic Tory MPs and Northern Ireland's Democratic Unionist Party".
Underlining the economic importance of Brexit, the head of British-headquartered pharmaceutical maker AstraZeneca said it would keep its freeze on manufacturing investments in the country unless a deal provides clarity about future ties.
The transition extension would nearly certainly ensure that the Brussels backstop proposal, where Northern Ireland remains within the EU's customs union and single market after Brexit, will never be enacted.
May insists any customs arrangement as part of the backstop must be temporary, ending at the latest in December 2021, but the European Union has refused to set an end date.