Fellow Americans Bethanie Mattek-Sands and Jack Sock downed Williams and Ram 6-7 (3/7), 6-1, 10-7 to grab the gold.
Venus won singles gold at Sydney in 2000 and doubles titles with sister Serena in 2000, 2008 and 2012. She and 32-year-old Ram had never played together before Rio.
"You always go into a match trying to win it and you never know if you are going to be able to cross that line," said 36-year-old Venus who had to be content with a record-equalling fifth medal in total.
The silver medal made up for a disappointing Games for the Williams sisters.
They lost in the first round of the doubles while Venus was also dumped out in the singles at the first hurdle.
Defending singles champion Serena was defeated in the third round.
Ram was a late call-up to the US squad after 2012 men's doubles champions Bob and Mike Bryan pulled out for health reasons.
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