The two were involved in a heated exchange during the second half of Madrid's 3-0 victory at 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.
"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."