May's special adviser Fiona Cunningham has also resigned, Downing Street has said.
The Tory Cabinet minister had clashed over ways to tackle extremism after reports of a hardline Muslim plot in some Birmingham schools. The Prime Minister has vowed to "sort out" the row, and asked Cabinet Secretary Sir Jeremy Heywood to investigate.
Gove also apologised to Home Office counter terror chief Charles Farr.
The dispute emerged when those close to Gove briefed the press about his frustrations that the Home Office was being too soft on extremism.