An increasingly desperate Theresa May on Sunday appeals to the British people to unite behind her Brexit deal as she calls on leavers and remainers to end hostilities and use the UK’s departure from the European Union to usher in a period of national “renewal and reconciliation”.

With criticism of the package negotiated by the prime minister and EU leaders continuing to grow, and a new row erupting on Saturday night over Gibraltar, May attempts to go over the heads of warring politicians in her own party by publishing a letter directly “to the British people”.

In a bid to persuade the public to apply pressure to their own MPs to back her deal in a crucial vote in parliament next month, the prime minister says in the letter that the country must “get on with Brexit now” so that ministers can focus on what matters most to people in their everyday lives – improving the NHS, building more homes and tackling injustices.

The letter has been distributed to the media and published on a new government website.

Promising to campaign for the deal with her “heart and soul” before the crucial Commons vote next month, May says Brexit day, on 29 March next year, “must mark the point when we put aside the labels of ‘leave’ and ‘remain’ for good and we come together again as one people”.

But as she headed to Brussels on Saturday night for a summit at which the 27 EU leaders are expected to sign off on both the withdrawal agreement and a separate political declaration, outlining the UK’s future relationship with the EU, senior Conservatives from both the leave and remain wings of her party appeared to be hardening their opposition to her plans and vowing to vote it down.

Several former cabinet ministers who served under the prime minister said she would have to go back to Brussels to renegotiate the package, as it would never pass through the Commons in its current form.

Speaking at the conference of the Democratic Unionist party in Belfast, which props up May’s government, the former foreign secretary Boris Johnson said the UK would become “a satellite state” under May’s plans and called for the backstop agreement, tortuously negotiated as a way to prevent a hard border between Northern Ireland and the Republic, to be “junked”.

“If we are not careful, we are going to stay in the customs union, we are going to stay in the single market, we are going to be rule-takers,” Johnson said. “Unless we junk this backstop, we will find that Brussels has got us exactly where they want us – a satellite state.”

The former Brexit secretary David Davis said he could not vote for the deal, which would leave the UK tied to the EU’s customs union and with no unilateral power to decide when to leave it. May’s deal, he said, had “a near-zero chance of getting through the House of Commons”, adding that the options were now “new deal or no deal”.

May was criticised on Saturday night by Brexiters for agreeing to Spanish demands regarding the future of Gibraltar, after Spain’s prime minister, Pedro Sánchez, threatened to veto the Brexit deal.

Britain’s ambassador to the EU, Sir Tim Barrow, made clear that Gibraltar would not necessarily be covered by a future trade deal with the EU – a key demand that they suggest will not now be met by the UK side. Stewart Jackson, former chief of staff to Davis, said the “Gibraltar surrender” could end up being “the straw that breaks the camel’s back”. However, May insisted on Saturday night that issues around Gibraltar’s sovereignty had not changed and would not change.

The prime minister has acceded to Spain’s demand that Gibraltar will not necessarily be covered by a future trade deal with the EU. Photograph: Emilio Morenatti/AP

Strong support for May came from her de facto deputy, the Cabinet Office minister David Lidington, who warned in an interview with the Observer that failure to back May could have catastrophic consequences. He said if there were a no-deal outcome resulting from MPs voting down the deal, the unity of the United Kingdom would be threatened and a “profound economic shock” would be triggered. Lidington said that “part of the political reality” was that “Scotland and Northern Ireland voted to remain in the European Union”.

Lidington dismissed alternative plans put forward by Johnson and the prominent Eurosceptic Jacob Rees-Mogg for a Canada-style deal that would see Northern Ireland treated differently from the rest of the UK, saying it was not on the table.

“The idea that there is some alternative that was capable of being negotiated is simply not the case,” he insisted. Lidington added: “Every member of parliament, whichever political party they represent, has got to really think through where the interest of the country lies when we come to the meaningful vote on the deal. A vote against the deal would be a vote to prolong and make worse the uncertainty for business. A vote against the deal would be to put at risk living standards and security of employment and investment in our own country.”

After Spain won the reassurances over Gibraltar, the European council’s president, Donald Tusk, said he would recommend that the EU approve the deal. “No one has reasons to be happy. But at least at this critical time the EU27 has passed the test of unity and solidarity,” he said.

If the deal is approved by the EU on Sunday, May will spend the next fortnight campaigning for it up and down the country before seeking parliament’s approval, probably in the second week of December.

But with Labour and up to 85 Tories saying they are likely to vote against, few see how she can avoid going down to defeat. The former cabinet minister Justine Greening, who backs a second referendum, united with Brexiters in condemning May’s plans. “This deal fixes nothing but risks everything. It leaves Britain’s economy and the union in a permanently weak position,” Greening said.