Just over 12 hours after May announced that her cabinet had agreed to the terms of the deal, Brexit minister Dominic Raab and work and pensions minister Esther McVey resigned on Thursday (local time).
Eurosceptics in May's Conservative Party said they had submitted letters calling for a vote of no confidence in her leadership.
"Am I going to see this through? Yes," May told reporters at her Downing Street office.
Two junior ministers, two ministerial aides and the Conservatives' vice chairman also quit. Hostility to the deal from government and opposition lawmakers raised the risk that the deal would be rejected and Britain would leave the EU on March 29 without a safety net.
May said she was sorry at the resignations and understood their unhappiness, but believed her deal was the right one.
"I believe with every fibre of my being that the course I have set out is the right one for our country and all our people," she said.
"I am going to my job of getting the best deal for Britain and I'm going to my job of getting a deal that is in the national interest."
May said no-one had produced an alternative to her Brexit plan, and that all options involved a backstop arrangement that has been criticised by many in her own party.
"Nobody has produced any alternative proposal," she told media, adding that repudiating a backstop altogether would kill any chance of a deal with the EU.
"If we do not move forward with that agreement, nobody can know for sure the consequences that will follow. It would be to take a path of deep and grave uncertainty, when the British people just want us to get on with it."
IRELAND 'WANTS MAY REPLACED'
By seeking to preserve the closest possible ties with the EU, May has upset her party's many advocates of a clean break, and Northern Ireland's Democratic Unionist Party (DUP), which props up her minority government.
The Daily Telegraph is reporting that the Conservative Party's deal with the DUP is over unless May was replaced with a new leader.
The DUP would vote down the withdrawal agreement in Parliament, adding that its support now "depended on who the leader of the Conservative Party is," the newspaper reported, citing sources close to DUP leader Arlene Foster.
Meanwhile, proponents of closer relations with the EU in her own party and the Labour opposition say the deal squanders the advantages of membership for little gain.
Both sides say it effectively cedes power to the EU without securing the promised benefits of greater autonomy.
"It is ... mathematically impossible to get this deal through the House of Commons. The stark reality is that it was dead on arrival," said Conservative Brexit-supporting lawmaker Mark Francois.
May will need the backing of about 320 lawmakers in the 650-seat parliament to pass the deal.
The ultimate outcome remains uncertain. Scenarios include May's deal ultimately winning approval; May losing her job; Britain leaving the bloc with no agreement; or even another referendum.
SIGN LANGUAGE VIDEO GOES VIRAL
A hilarious video has emerged showing the sign language interpreter for BBC News displaying amazing facial expressions as she tries to translate what is going on in Brexit.
POUND PLUNGES
Business leaders expressed growing alarm as the draft Brexit agreement - seen as the only chance of preserving some stability in UK-EU trading threatened to unravel - sent stock prices and the pound plunging on Thursday.
The European Union is Britain's biggest trading partner, accounting for 44% of UK exports and 53% of imports to the United Kingdom.
After 45 years of membership, industries including defence, cars and aerospace have created intricate supply chains that rely on smooth, "just-in-time" delivery of thousands of parts across the sea that divides Britain from the continent.
Business leaders fear that the country could stumble towards a no-deal Brexit where border checks block ports and fracture the supply chains that support the likes of Rolls-Royce and BAE Systems .
A senior executive at one of Britain's biggest banks said this was the most disastrous government he had ever seen.
