In a statement outside Downing Street, he said he would attempt to "steady the ship" over the coming weeks and months but that "fresh leadership" was needed.
The PM had urged the country to vote Remain but but was defeated by 52% to 48% despite London, Scotland and Northern Ireland backing staying in.
UKIP leader Nigel Farage hailed it as the UK's "independence day".
The pound fell to its lowest level against the dollar since 1985 as the markets reacted to the results.
