Fulham complained after Liverpool's website announced in late August that Dempsey had signed for the club, and published subsequent praise for the Texan from manager Brendan Rodgers. Fulham believe Liverpool's actions unsettled Dempsey and prompted his deadline-day move to Tottenham.
In a climbdown that will increase scepticism over the conduct of Liverpool's owners, the Fenway Sports Group, Werner told Al Fayed the Reds' behaviour had been wrong.
Werner's apology was followed by a letter from Anfield managing director Ian Ayre to his Fulham counterpart, Alistair Mackintosh.
In the letter Ayre acknowledged Liverpool's conduct was "wrong", "should simply never have occurred", and expressed "sincere regret and apology".
Ayre is understood to have said Liverpool "can do better" and "pledged" to try and do so. He also praised Fayed's leadership, thanked him for putting "the best interests of the game" ahead of the dispute, and called him "magnanimous" for withdrawing the complaint.
Despite the apology, and Fulham's confirmation yesterday they had withdrawn the complaint, the Premier League board will still consider the allegation of a rule breach made by Fulham.