The two were involved in a heated exchange during the second half of Madrid's 3-0 victory at the Coliseum Alfonso Perez.
It was alleged on social media that Cucurella, who joined Getafe on a season-long loan from Barcelona last July, addressed Militao with the word 'mico', which means monkey in Catalan.
Cucurella said he used the Castilian Spanish term 'pico', which translates in the context used as mouth.
Me gustaría aclarar que en ningún momento he insultado ni descalificado a Militao. Mis palabras han sido "Cierra el p... pico".— Marc Cucurella (@cucurella3) January 4, 2020
Pido perdón si alguien se ha sentido ofendido por ello. Estas son las únicas declaraciones que he hecho y que voy a hacer al respecto.
"I would like to clarify that in no moment did I insult Militao. My words were 'Shut your f****** mouth'," he posted on Twitter.
"I apologise if anyone has been offended by this. These are the only statements I have made and that I will make about it. Thank you."