Philip Christian, 68, said in the 1970s that he penned the words and melody that became the main theme from the hit 1981 show Cats.
He alleged that a member of one of Lord Lloyd-Webber’s other works overheard a recording of the song and recited it to him ahead of the show’s production.
Mr Christian was suing Lord Lloyd-Webber and lyricist Sir Trevor Nunn for breach of copyright.
He claimed he owns the rights to the “lyrics and musical score” and demanded future royalty payments.
But following a High Court hearing in London last week, judge Master James Brightwell threw out the “fanciful and entirely hopeless” claim.
Rights to the song Memory were registered to Lord Lloyd-Webber and to Sir Trevor, who shares the copyright in the words with the family of poet TS Eliot, whose work Rhapsody on a Windy Night was the inspiration.
Mr Christian told the court his own song was inspired by his experience arriving in the UK aged 11 from the Dominican Republic in the 1960s.
He said he was training in performance arts when a recording of his song was heard by one of Lord Lloyd-Webber’s dancers.
He claimed it must have been “memorised and then recited” in front of the star composer.
Representing himself, Mr Christian said: “I have always had that tune in my head. Every time I hear it on the radio, I get angrier.”
Barrister Stephanie Wickenden, however, argued his copyright claim should be struck out.
She said his argument the song was memorised then recited was “not a plausible allegation”, adding: “If there were any merit to the claim, it would’ve been brought 40 years ago.”
The judge said Mr Christian had failed to produce a copy of the song.
He added: “I find it inherently incredible that a dancer could, in the context of the entrance hall to a dance studio, have remembered word for word both the lyrics of the song and the melody and been able to communicate them at a later point.”
He granted summary judgment against Mr Christian.