The 25-year-old journalist, model, television presenter and mother-of-two died from an overdose at her home in Wrotham, Kent, on April 7, 2014.
Geldof had been a heroin addict and took the substitute drug methadone for two years before her death.
The Kent Police has been probing the case for the last 15 months, but no arrests have been made and officers said all lines of inquiry have now been exhausted.
Fotheringham said the case would be reviewed if further information or witnesses came forward.
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"I would like to take this opportunity to pay tribute to Peaches' family who have supported our inquiries during what has been a difficult time," he said.
A police search of her home at the time of her death found 6.9 grams of heroin but there was no indication she had planned to take her own life, an inquest last year heard.
The musician, who had been away for the weekend with the eldest of their two sons, said he became concerned when she failed to answer her phone and found his wife slumped on the bed upon his return.
Coroner Roger Hatch said Geldof's death had been "drugs related" and expressed his deep sympathy to her family.
Her mother, Paula Yates, died of a heroin overdose at the age of 41 when Peaches was 11 years old.
In a television interview last year, Geldof's father, Sir Bob Geldof, the lead singer with the Boomtown Rats and noted anti-poverty campaigner, said he "blames himself" for her death, saying: "You're the father who is responsible and clearly failed.