The Ultravox frontman, who co-organised the Band Aid, Live Aid and Live 8 fundraisers with her father Sir Bob Geldof, said he was stunned by the tragedy.
Ure said: "I think disbelief is the first sensation you get and when you check it on the Internet and television, I think the horror is just astounding. She was great, she was a bouncy, lovely child... (her mother Paula Yates) was this 'earth mother' and that's exactly what Peaches has turned into. It's uncanny. The fact that Peaches posted (a) photograph of Paula and her together the day before it happened, and the fact that they embraced this whole motherhood thing so incredibly well after what had been maybe ropey beginnings, they had completely found their niche in life. A tragedy to happen twice in slightly different circumstances is just beyond belief."
Ms Geldof was found dead at her home in Wrotham, Kent, on Monday after officers were called "following a report of concern for the welfare of a woman".
A post-mortem examination is due to take place today. Police said they were treating the death as sudden and unexplained but not suspicious. There was reportedly no suicide note or illicit drugs found at the home which Ms Geldof shared with her husband, Tom Cohen and their two infant sons.
Yates died of an accidental heroin overdose in 2000, three years after the apparent suicide of her partner, Michael Hutchence. She had left Sir Bob for the INXS star in 1995, sparking an acrimonious divorce. However, Sir Bob went on to adopt the couple's only child, Tiger Lily.
Ms Geldof's elder sister, Fifi Trixibelle, also paid tribute. Posting a picture on Instagram of the two of them together as children, she wrote: "My beautiful baby sister.... Gone but never forgotten."
A coroner will open an investigation after the results of the initial post mortem examination have been received. A decision on whether to hold an inquest will be made depending on the final results, which could take weeks.