In a letter to Prime Minister Theresa May, Duncan said he wants to be "free to express my views in advance of you relinquishing office." May said she would step down as prime minister after three failed attempts at a deal for Britain to leave the European Union. Duncan said May "deserved better."
"The U.K. does so much good in the world," Duncan said. "It is tragic that just when we could have been the dominant intellectual and political force throughout Europe, and beyond, we have had to spend every day working beneath the dark cloud of Brexit."
Duncan has criticized Johnson in the past, calling him a "circus act." Last year, he called for the "political end of Boris Johnson."
Their most recent spat was over U.S. Ambassador to Britain Kim Darroch, who resigned this month after diplomatic cables were leaked in which he called President Donald Trump a "pompous fool" and labeled his administration as "utterly dysfunctional." Duncan said Johnson threw Darroch "under the bus" by forcing him out.
Chancellor Philip Hammond and Justice Secretary David Gauke announced they would resign this week, as well.
Britain will learn who will succeed May this week after the Conservative Party voting ends. The winner is expected to be announced Tuesday and take office Wednesday.
Johnson supports London's EU departure and has said Britain must leave "do or die, come what may" on Oct. 31, with or without a deal. His primary opponent, Jeremy Hunt, initially opposed leaving in the 2016 referendum but now strongly supports leaving with a deal.