Cameroon international Bong said the incident took place during the Baggies' 2-0 win at The Hawthorns on 13 January.
Rodriguez, who has one England cap, appeared to pinch his nose after they clashed and Bong spoke to the referee.
In his evidence to the FA, Bong said he was "100% certain" Rodriguez, 28, said to him: "You're black and you stink."
Rodriguez, who had a personal hearing with the governing body on Wednesday, denied saying this, insisting it had been: "Breath fucking stinks."
The FA employed two lip-reading experts to watch slow-motion footage of the incident.
In the reasons explaining its verdict, the FA said the "essential issue for us boiled down to one question - are we satisfied the player [Rodriguez] probably said to GB [Gaetan Bong]: "You're black and you stink'?
It said the two lip-reading experts "could not help" on this "core issue".
"The player's mouth was obscured, and neither could see sufficient to interpret his moving lips," the FA said.
"After much deliberation we were left in the position where the case distilled to the evidence of each player. We could not say that any of the other evidence or competing arguments lead us to prefer one over the other."
The FA said it was "completely satisfied" 29-year-old Bong's complaint was "made in absolute good faith" and "there has been no suggestion that this was a malicious or fabricated complaint".
It said it "takes all allegations of discrimination extremely seriously and continues to encourage all participants who believe they have been the subject of or witness to discriminatory abuse to report this through the appropriate channels".
Had he been found guilty of the charge, Rodriguez would have received a minimum five-match suspension.
During his hearing, Rodriguez provided character evidence in the form of statements and letters from 11 people, including Tottenham manager Mauricio Pochettino, Bournemouth manager Eddie Howe, Liverpool midfielder Adam Lallana, Southampton striker Shane Long, defender Maya Yoshida and former Saints keeper Kelvin Davis.
West Brom director of football administration Richard Garlick, who attended the hearing, said: "Everyone at the club is delighted for Jay because this has been a trying period for him."
Bong said he was "very disappointed" by the FA's verdict and that "despite the lack of evidence" he remains "certain of what I heard on the field".
"I have been in this country for almost three years and this is the first time that such a thing has happened to me," a statement read.
"But in spite of my disappointment I encourage everyone who may also be subjected to a similar experience to stand up and denounce such acts, even if the chances of obtaining sanctions are likely to be very small.
"Human rights are universal rights that are not foreign to any culture and they belong to all countries!" AGENCIES