Rebel elite soldiers overthrew president Mohamed Bazoum on July 26 and have since detained him at home with his family.
Negotiations to restore civilian rule have yet to bear fruit, with the coup leaders demanding a three-year transition and the Economic Community of West African States (ECOWAS) calling for the immediate return of the democratically elected Bazoum.
The coup has also been strongly criticized by Western governments and global bodies such as the UN, which is holding its General Assembly of world leaders in New York this week.
In a news release read on public television, the military said UN Secretary-General Antonio Guterres "went astray in the exercise of his mission by obstructing Niger's full participation in the 78th session of the UN General Assembly".
It criticized "the perfidious actions" of the UN leader, adding that they were "likely to undermine any effort to end the crisis in our country."
Bakary Yaou Sangare, who before the coup was Niger's ambassador to the UN and is now its foreign minister, was the new leaders' chosen representative for the gathering. But, according to a diplomatic source, there was also an application by the overthrown government to represent Niamey.
"In case of competing credentials from a Member State the secretary-general defers the matter to the Credentials Committee of the General Assembly who will deliberate on the matter," Guterres' spokesman Stephane Dujarric said. "The secretary-general does not decide."
Because the committee will not meet until later, no representative from Niger was added to the speakers' list.
Niger "forcefully rejects and denounces this clear interference by Mr Guterres in the internal affairs of a sovereign state", the country's leadership said.
One of the world's poorest nations, Niger is the fourth country in West Africa to suffer a coup since 2020, following Burkina Faso, Guinea and Mali. Bazoum's removal heightened international worries over the Sahel region, which faces growing jihadist insurgencies linked to Al-Qaeda* and the Islamic State group*.
Regional sanctions since the coup mean food and medicines are scarce in landlocked Niger, prices are skyrocketing and there are blackouts after Nigeria cut electricity supplies.
Senegal's President Macky Sall said on Thursday a diplomatic solution in Niger was "still possible".
"I hope that reason will ultimately prevail... that it is still possible to move forward reasonably to a solution," Sall said in an interview with France's media outlets.
He urged Niger's coup leaders "to not push (us) to the final decision which would be a military intervention."
The military leaders of Mali, Burkina Faso and Niger signed a mutual defense pact this month, saying they aimed to "establish an architecture of collective defense and mutual assistance for the benefit of our populations."
* The Islamic State (IS)/Daesh and al-Qaeda are terrorist organizations banned in Russia and many other countries around the world.