Mr Davies said he would consult the Welsh Labour Party before making a decision on his candidacy for the post, but it is highly unlikely that he will run.
The new Secretary of State for Wales, Alun Michael, is not even a candidate for next year's Assembly elections and has made no secret of his desire for much higher office.
The most obvious replacement as Labour's candidate for the First Minister's job is Rhodri Morgan, MP for Cardiff West and chairman of the powerful Commons Public Administration Select Committee.
Mr Morgan, who said last night he was "too stunned" by the news to make a statement, won the support of more than half of Labour's constituency parties and several MPs when he was beaten by Mr Davies in the selection contest last month.
If Mr Davies does decide against standing, Labour will almost certainly have to re-run the contest, and Mr Morgan is sure to be the front-runner. However, the party leadership may prefer a more malleable figure, and speculation last night centred on Peter Hain, a Welsh Office minister and MP for Neath, and Paul Murphy, a Northern Ireland minister and MP for Torfaen.
The Conservatives' constitutional spokesman, Liam Fox, said it was inconceivable that Mr Davies could continue as party leader in Wales. "His resignation means that he has to think again about his bid to be leader of the Welsh Assembly. It would be utterly unacceptable for him to stand."
Nigel Evans, the Conservative spokesman on Welsh Affairs, added: "If the error of judgement is sufficiently serious for him not to wish to embarrass the Labour Party in Westminster, then I assume that he would not wish to embarrass the people of Wales by remaining as leader of the Welsh Labour Party."
The leader of Plaid Cymru, Dafydd Wigley, said the issue of the Labour candidacy "had to be answered", though he stressed that Mr Davies's resignation was a "blow to Wales".
Welsh MPs who have been critical of Mr Davies's leadership rallied round last night, claiming that his family should be protected from further speculation.
Llew Smith, MP for Blaenau Gwent and leader of the "no" campaign in the Welsh referendum, said: "We have had our political differences, but no one would wish this on anyone. It is a tragedy."
Denzil Davies, a former shadow secretary of state for Wales, said: "It is very, very sad for him and his family. He was damned good at his job."
Paul Boateng, formerly a Health minister, has replaced Mr Michael as Home Office minister. Mr Boateng was in turn replaced by John Hutton, MP for Barrow-in-Furness and parliamentary private secretary to the Leader of the House, Margaret Beckett.