Britain's Laurence Godfrey was beaten 113-112 by Australia's Tim Cuddihy in the men's individual 70m archery bronze medal play-off.
The 28-year-old Briton was aiming to emulate Alison Williamson's success in the women's competition on Wednesday.
Despite a late Godfrey comeback in the final end, Cuddihy fired an eight to end Godfrey's hopes of bronze.
Italy's Marco Galiazzo, who beat Godfrey in the last four, claimed gold when he beat Hiroshi Yamamoto.
European champion Galiazzo edged out his Japanese opponent 111-109 in a tense final under the sorching sun in the Panathinkaiko Stadium.
Despite his near medal miss, Godfrey was in upbeat mood.
"I am happy with anything," said the British champion. "All I expected was to give 100% and do my best and it gave me fourth.
"If giving 100% had given me 64th then so be it, but it has given me fourth and that is fantastic."
Godfrey was trailing from the very first arrow sent down by his 17-year-old opponent from Canberra.
Cuddihy, who lost to Yamamoto in a tense semi-final, dropped only three points from his first nine arrows.