The Cottagers reported Liverpool to the Premier League this summer after becoming angered with their public attempts to sign the midfielder.
Despite Dempsey joining Tottenham, Fulham initially decided to press ahead with their complaint.
But they have now dropped the matter after Liverpool chairman Tom Werner apologised to his Fulham counterpart Mohamed Al Fayed.
A statement on Fulham's website read: "Fulham can confirm that it has today withdrawn the complaint made against Liverpool FC in regard to the Merseyside club's inappropriate actions surrounding interest in Clint Dempsey over the summer.
"Liverpool's chairman Tom Werner apologised personally to Mohamed Al Fayed, stating that his club were wholly responsible for unprofessional behaviour toward Fulham and emphasised his regret for the entire episode.
"In a letter from Liverpool to Fulham's CEO Alistair Mackintosh, Tom Werner's praise of the magnanimous gesture from the chairman was reiterated, as was the club's assurance that the conduct of Liverpool was simply not acceptable.
"The chairman decided that this sincere action was enough, and as a gesture of goodwill, instructed that all complaints against them be dropped."