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.

Junction One and The Outlet are for sale with a combined guide price of between £58m and £62m, according to property consultants CBRE.
The Outlet opened on the outskirts of Banbridge in County Down in 2007, but was effectively taken over by Ulster Bank in 2011.
Junction One opened near Antrim in 2004.
Ulster Bank also lent the money for that development.
The Outlet is currently held by an Ulster Bank company called West Register (Northern Ireland) Property.
It is a Northern Ireland branch of a group of companies set up by the bank's parent company RBS to hold "distressed properties".
Earlier this year another West Register firm sold the Richmond Centre in Londonderry.
The Outlet was opened in 2007 at a cost of £70m but struggled to fill all its units.
There are plans to switch part of its use from retail to leisure with a cinema and restaurants.
The last set of accounts for the company that holds Junction One valued the development at £7.85m, compared to a valuation of almost £80m in 2009.
Junction One Ltd had Ulster Bank borrowings of about £48m and its liabilities outweigh its assets by £53m.
Leona Barr, centre manager for Junction One, said: "We welcome that the Junction One complex is on the market as it brings great potential for future development and continued investment.
"We believe that the time is right for new owners to build on the success already achieved and to input additional investment to enable the complex to reach its full potential."

Dylann Roof was convicted last month of 33 federal charges, including hate crimes, after opening fire on a Bible study group in 2015.
He remained unrepentant and had told the jury: "I felt like I had to do it and I still feel like I had to do it."
The jurors deliberated for nearly three hours before reaching their verdict.
The massacre shocked the nation and reignited a debate about race relations and the flying of the Confederate flag.
Roof told police he wanted to start a race war and he was photographed holding the battle flag, which to many is a symbol of hate.
The tragedy led to the flag being removed from the South Carolina statehouse, where it had flown for 50 years.
Earlier on Tuesday, Roof addressed the jury, saying: "I don't know what good it would do anyway" if they spared his life.
The judge will issue a formal sentence on Wednesday morning.
After the sentence was announced, the 22-year-old killer stood and requested to be appointed new lawyers and to file for a retrial.
US District Judge Richard Gergel replied that he was "strongly disinclined" and instructed Roof to think about it overnight.
Nearly two dozen friends and relatives of those gunned down at the Emmanuel AME Church in Charleston testified during the sentencing phase about how Roof's crimes have affected their lives.
But none of them had appealed to the jury to return a death sentence.
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.
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Two major out-of-town retail centres that struggled in the economic downturn have been placed on the market.
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A white supremacist has been sentenced to death for the racially motivated killings of nine black people at a South Carolina church.