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".