She previously held the role of culture secretary, until Prime Minister Theresa May reshuffled her cabinet on Monday.
But he said he recognised that he would not "be able to give the effort, energy and complete focus needed at this important time" and therefore believed that "the right thing at this time is for me to stand down from my current responsibilities".
It's now one year since the power sharing executive in Northern Ireland collapsed after Sinn Féin walked out on government, citing a number of key issues with their partners the DUP.
The business group also wished James Brokenshire a speedy recovery and all the very best for the future.
Mr Brokenshire's resignation came prior to the Prime Minister's cabinet reshuffle.
Asked for her reaction to the postponement of the meeting between Mrs Bradley and Mrs O'Neill, the DUP leader said: "It's another indication that all is not well in Sinn Fein and they are a party in disarray at present".
Mr Coveney is now on a visit to the Middle East and will stop over in London on his way back to Ireland.More news: Friends of Home and Away's Jessica Falkholt held vigil
"His unwavering commitment - in public and in private - over the previous year to securing the effective operation of the devolved power-sharing institutions in Belfast has been hugely important", he said.
It is apt that the word likely to be most associated with James Brokesnhire's year and a half as Northern Ireland secretary is "deadlines".
In a statement, the Tánaiste and Minister for Foreign Affairs and Trade, Simon Coveney said: "I learned with regret today that James Brokenshire has resigned as Secretary of State for Northern Ireland for health reasons".
Following a meeting with Conservative leader May, 50-year-old James Brokenshire, whose job had been to represent Northern Ireland in the United Kingdom cabinet and oversee, announced he was to step down ahead of major lung surgery, Efe news agency reported.
In November, Brokenshire was forced to begin setting a budget for the province, but he said a return to direct rule could still be avoided.
"This is clearly a hard time for not only James but for Cathy, his wife and his children".
"I think an bad lot has happened in the year - two elections - we've had a very clear mandate in relation to the way forward".