Coty had first made a public offer for Avon in April, but later told that company's board that it had until the close of business on Monday to start talks.
Your total lack of engagement with us leads us to believe that you remain reluctant to explore a friendly, negotiated combination on a reasonable timetable, Coty Chairman Bart Becht said in a letter to Avon dated Monday and made public.
It is time for Coty Inc to move on and pursue other opportunities.
Avon did not immediately respond to requests for comment.
Shares of Avon on Monday closed up 77 cents at $20.96 on the New York Stock Exchange. Coty revealed its decision after U.S. markets closed.
The stock rose after Avon said on Sunday that it would consider the takeover bid by its smaller rival, but that it might need a week to respond.
Coty, the maker of Stetson aftershave and Beyoncé fragrances, first reached out to Avon in March, and later made public its offer to buy the company for $23.25 per share.
That offer included financial backing from Warren Buffett's Berkshire Hathaway Inc.
Although it has close to three times as much revenue as Coty, Avon faces declining sales at home, as well as a U.S. probe into overseas bribery allegations.
(Reporting By Phil Wahba and Jonathan Stempel in New York; Editing by John Mair and Edmund Klamann)