Kevin Prince Boateng and Sulley Muntari were thrown out of the squad on the eve of a decisive match against Portugal and Boateng claims there was financial mismanagement and "amateurish" organisation.
Coach Kwesi Appiah then told media that Essien refused to take part in a warm-up session ahead of that game, so would have to face punishment.
"I have monitored with great concern, the raging public and media debate about the reasons for the Ghana Black Stars' early exit from the ongoing tournament in Brazil," declared Essien in a statement on Citi FM radio.
"Like any other patriotic Ghanaian, and as an avid supporter and member of the Ghana Black Stars, I am deeply embarrassed and devastated by the team's failure to make, at least, Semi Final berth we had publicly promised the good people of Ghana ahead of our departure from Accra.
"While I will be the first to concede that not everything went according to plan before and during our participation in the tournament, I believe that destructive and vicious attacks and counter attacks on individual members of the team as well as the technical and management team are not the solution to the Ghana Black Stars' problems.
"Instead, we must all embrace peace and seek a lasting solution for the team's problems through decent dialogue.
"I believe that building genuine cooperation among all stakeholders in Ghana Football remains essential for promotion of trust, transparency, and good governance to ensure prosperity of the senior national team, as well as the sport in general.
"While I believe we desperately need a peaceful resolution of all the issues agitating our minds and the minds of all fans of the team, I hold the view that we can investigate and resolve these issues without further destroying the team."
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