Holgate accused Firmino of using a discriminatory term towards him after the two players clashed in the third-round encounter at Anfield, which finished 2-1 to the hosts.
The Football Association has investigated the incident and, in a statement released on Wednesday, it confirmed that the Reds forward will not face any action.
In his own statement, Firmino has stressed that the claims were false and that he understands the severity of racist abuse on the football field as he has also suffered from it in the past.
"As difficult as it has been to remain publicly silent, given the serious and damaging nature of what it was claimed I said during the game, I did so to demonstrate my respect for the process and to allow the issue to be investigated in the most thorough way," the Brazilian wrote.
"It is critical for football that tackling racism and all forms of discrimination is taken extremely seriously. As someone who has experienced racist abuse during my life, I know how damaging and hurtful it can be.
"Now the process is concluded, I would like to place on record, for the avoidance of any doubt, I did not say the word, or a variation of the word, that was claimed and subsequently reported in the media. I did not use any language that referenced race. I did not - and would never - reference a person's skin colour or culture, by means of insult, during a dispute or an argument.
"There is no place for discrimination on a football pitch, or anywhere else in life for that matter. I am pleased after exhaustive reviews of all the evidence, in this specific incident, the matter is resolved."
Everton confirmed that they would accept the findings of the investigation.