The Leave-backing minister said he wanted to return to the backbenches "to be free to participate in the critical debate that will take place in the weeks ahead".
He said Mrs May had undertaken a series of "undignified retreats" on Brexit and warned that prolonging the two-year Article 50 negotiation process could result in "the final humiliation of our country", with the EU dictating the terms of any extension.
The Camborne and Redruth MP said he would back Mrs May's Withdrawal Agreement when it returns for another "meaningful vote" in the Commons, due by March 12.
But he made clear he was ready to support a no-deal Brexit if the agreement is rejected by MPs.
Britain needs to be "ready to face down the European Union here and now" , he said in his resignation letter to the Prime Minister.
"We cannot negotiate a successful Brexit unless we are prepared to walk through the door."
Mr Eustice was among Conservative ministers who voted in favour of an amendment on Wednesday paving the way for a potential vote to delay Brexit.
The Prime Minister's decision to whip her MPs to support the amendment caused discontent among Leave-backing Tories, with 20 rebelling to vote against and scores abstaining.
It is understood that Mr Eustice spoke briefly by phone with Mrs May on Thursday afternoon about his decision to quit.
In a letter responding to his resignation, Mrs May said: "I agree with you that Parliament must now come together and honour the referendum result by voting for a deal which will give businesses and citizens the certainty they need and deserve.
"Our absolute focus should be on getting a deal that can command support in Parliament on March 29. It is within our grasp and I am grateful to have your continued support in that important mission."
Before becoming a Conservative MP, Mr Eustice stood as a Ukip candidate in the 1999 European Parliament election and led the No campaign against the UK joining the euro.
A press officer for Michael Howard and David Cameron in opposition before his election to Parliament in 2010, Mr Eustice joined the Department for Environment, Food and Rural Affairs in 2013 and had held his latest post since 2015.
He campaigned to leave the EU in the 2016 referendum and backed the approach set out by Mrs May at Chequers last year, when a number of Brexit-backing ministers resigned.
In his letter of resignation, he voiced concern that Mrs May had handed MPs "direct control of events" by announcing that if her Agreement is defeated, she will offer them a vote on whether to rule out a no-deal departure from the EU on March 29 and whether to request an extension of talks.
"I have stuck with the Government through a series of rather undignified retreats," said Mr Eustice.
"However, I fear that developments this week will lead to a sequence of events culminating in the EU dictating the terms of any extension requested and the final humiliation of our country.
"I appreciate that you have been terribly undermined by those in Parliament who refuse to respect the referendum result. You have shown tenacity and resilience over the past year.
"However, what our country needs from all its political leaders at this critical juncture is courage, and we are about to find out whether Parliament has it."
Mr Eustice said he did not believe the European Commission had behaved "honourably" during the two years of negotiations since the UK notified Brussels of its intention to quit in 2017.
He said: "They have deliberately made progress slow and difficult. They have stated in terms that they will refuse to even hold substantive negotiations on a future partnership until after we leave.
"If the position of Parliament is now that we will refuse to leave without an agreement then we are somewhat stuck.
"This is uncomfortable for everyone, but we cannot negotiate a successful Brexit unless we are prepared to walk through the door.
"We must therefore have the courage, if necessary, to reclaim our freedom first and talk afterwards. We must be ready to face down the European Union here and now.
"The absence of an agreement poses risks and costs for them too. We already know that in the event of no deal, the EU will seek an informal transition period for nine months in many areas and settlement talks could continue within this window.
"I will do what I can from the back benches to try to salvage this sorry situation and I hope that, when the moment comes, Parliament will not let our country down."
Mr Eustice won warm tributes from many Conservative MPs, including Environment Secretary Michael Gove, who said in a tweet: "So sorry to see George go. He has been a brilliant minister and will remain a dear friend.
"He leaves an outstanding legacy, with the Agriculture and Fisheries Bills setting domestic policy for the first time in nearly 50 years. He will be very much missed.
But the Scottish National Party's Westminster leader Ian Blackford said: "Another day another resignation from the UK Government.
"Any illusion to strong and stable ended before it began, but this is beyond parody. This is not a functioning Government it is the Tory Party at war with itself under a Prime Minister who can't give leadership."
The Country Land and Business Association (CLA) responded to Mr Eustice's resignation with disappointment.
CLA president Tim Breitmeyer said: “We are saddened to hear of George Eustice’s decision to resign as Farming Minister at the Department for Environment, Food and Rural Affairs. He has held the position since 2015, and has maintained a strong voice on behalf of the farming industry during that time.
“His farming background and first hand knowledge and experience have been invaluable in the many areas of his brief. The farming community has lost a key ally at this critical time for the industry, which faces significant uncertainty and change.”