Martin Jol's side reported Liverpool to the Premier League during the summer after they became annoyed at their approaches for the forward.
Fulham were angered by an article on the Fenway Sports Group website that claimed Dempsey joined the Anfield club.
Reds manager Brendan Rodgers did express an interest in signing Dempsey but no formal bid was lodged, prompting Jol to complain Liverpool had unsettled the forward with their "embarrassing" conduct.
Dempsey eventually joined Tottenham and Liverpool chairman Tom Werner has now apologised to his Fulham counterpart Mohamed Al Fayed to bring the matter to a close.
A statement on Fulham's website read: "Fulham can confirm that it has today withdrawn the complaint made against Liverpool FC in regard to the Merseyside club's inappropriate actions surrounding interest in Clint Dempsey over the summer.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."
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