Her decision to resign as International Development Secretary came after being summoned back from an official visit to Africa for a showdown with Theresa May in Downing Street.
The Prime Minister has now lost two Cabinet ministers in successive weeks and will face a politically sensitive reshuffle to replace the prominent Brexit-backer.
Ms Patel had been intending to spend three days in Kenya and Uganda, but was forced to cut short her trip and return home from Nairobi to explain the disclosure of further unauthorised meetings with Israeli politicians.
In a letter to the Prime Minister, Ms Patel said: "I offer a fulsome apology to you and to the Government for what has happened and offer my resignation."
In reply the Prime Minister told her that meetings should have been arranged "formally and through official channels".
Witham MP Ms Patel's downfall came after it emerged she had a series of 12 engagements with senior Israeli figures - including prime minister Benjamin Netanyahu - during a holiday in the country in August.
She then held two additional meetings, one in the UK and one in the US, following her return from Israel.
In a further development, the Israeli Haaretz newspaper reported that during her stay in the country she visited an Israeli military field hospital in the occupied Golan Heights.
Britain, like other members of the international community, has never recognised Israeli control of the area seized from Syria in the 1967 Six-Day War.
In her resignation letter Ms Patel said: "In recent days there have been a number of reports about my actions and I am sorry that these have served as a distraction from the work of the Department for International Development and of the Government as a whole.
"As you know from our discussions I accept that in meeting with organisations and politicians during a private holiday in Israel my actions fell below the high standards that are expected of a Secretary of State.
"While my actions were meant with the best of intentions, my actions also fell below the standards of transparency and openness that I have promoted and advocated."
The meetings, without officials and in relation to one of the most sensitive areas of foreign policy, led to Ms Patel apologising and being given a dressing down by Mrs May on Monday.
But subsequent disclosures added to pressure on Ms Patel, culminating in a meeting in Downing Street lasting around 30 minutes during which it was made clear her Cabinet career was over.
In her letter to Ms Patel, the Prime Minister said said: "As you know, the UK and Israel are close allies, and it is right that we should work closely together.
"But that must be done formally, and through official channels.
"That is why, when we met on Monday, I was glad to accept your apology and welcomed your clarification about the trip to Israel over the summer.
"Now that further details have come to light, it is right that you have decided to resign and adhere to the high standards of transparency and openness that you have advocated."
Under intense media scrutiny - including thousands of people following the progress of her plane on a flight tracking website - Ms Patel had arrived back in the UK on a Kenya Airways flight to Heathrow.
Her early return to the UK followed the disclosure that she met Israeli public security minister Gilad Erdan in Parliament on September 7, and foreign ministry official Yuval Rotem in New York on September 18.
It is understood that Downing Street was told about the New York breakfast with Mr Rotem when Ms Patel revealed the details of her trip to Israel on Friday, but No 10 only learnt on Tuesday about the meeting with Mr Erdan.
No British civil servants were present and, like her meetings in Israel, she did not report them to the Foreign Office or Government in the usual way.
She was accompanied at all the meetings bar one in Israel by the honorary president of the Conservative Friends of Israel lobbying group, Lord Polak.
Ms Patel's resignation comes a week after Sir Michael Fallon quit as defence secretary over allegations of inappropriate behaviour towards women.
The Prime Minister could be keen to replace Ms Patel with another Brexit-backer in order to avoid further discontent among pro-Leave MPs on the Conservative benches and maintain the current balance within the Cabinet.