Mr Nyantakyi has not replied to multiple requests for comment while the GFA has said it would cooperate with any investigation.
"I wish to clarify that at no time prior to or subsequent to any deliberations on the video did I consult, contact or inform any government official about discussions on the video. This is by no means an admission of wrongdoing for issues on the video", the statement said.
Kwesi Nyantakyi's apology and resignation come a day after the government announced a dissolution of the Ghana Football Association and hours after FIFA banned the embattled football president for 90 days.
Any FIFA ban could also affect the under-17 women's team, who are preparing for the U-17 World Cup; top club Aduana Stars, who are participating the CAF Confederation Cup; and the men's national team, who are due to play in three African Cup of Nations qualifiers later this year. However, parts of its rules is that governments should not interfere in football administration.
The GFA said it would cooperate with any investigation.
Nyantakyi was accused in an explosive documentary called "Number 12", aired on Wednesday by investigative journalist Anas Aremeyaw Anas, of soliciting bribes amounting to several million dollars.
Abdul-Hamid said the government was "shocked and outraged" at the claims, which included referees apparently accepting bribes of at least $100 to throw matches.
The Ministry say they have been forced to act "as a result of the pervasive nature of the rot within the GFA" and that they have "taken immediate steps to have the GFA dissolved".
A police vehicle is pictured at the premises of Ghana Football Association in Accra.
"I wish to apologise to my family, work colleagues, friends, associates and the people of Ghana for the disappointment my indiscretion has caused them".
Ghana missed out on the World Cup which starts in Russian Federation on June 14.
Black Maidens won bronze at the U-17 World Cup.