Williams met the Olympic qualifying time winning the IFAM title in Belgium in May but failed to gain automatic selection at the British Championships.
Sebastian Rodger won the British title with Jack Green second and subsequently both secured selection in Birmingham.
Williams, 32, ran at the 2012 Olympics and won European gold the same year.
"Rhys was only eligible for selection in round four of selection, and to be successful in this round the selection panel would need to believe he would win a medal in Rio or at a future Olympic Games and/or there were exceptional circumstances surrounding his potential selection/qualification," a British Athletics statement read before the appeal.
"Unfortunately it was felt that Rhys did not fit into either of these categories and was therefore not selected."
Williams, who received a four-month ban for doping in 2014, ran just under British Athletics' Olympic A standard, winning his Belgium meet in May in a time of 49.22 seconds. The British Olympic qualifying time for the 400m hurdles was 49.40 seconds.
Williams believed he should be eligible but British Athletics have rejected his appeal.
Seren Bundy-Davies, who helped Great Britain win 4x400m European gold on the weekend, is the only Welsh track and field athlete named in Great Britain's team for Rio 2016 meaning Wales will have just one track and field athlete at an Olympic Games for the first time since the 1952 Helsinki Games.
Welsh 400m hurdler Rhys Williams has failed in his appeal against his omission from Team GB's athletics squad for the Rio Olympics next month.