In an opinion piece in the Washington Post, President Mohamed Bazoum said he was writing “as a hostage”.
He also warned that the region could fall further under Russian influence, via the Wagner Group which already operates in neighbouring countries.
Niger’s West African neighbours have threatened military intervention.
On Thursday, the coup leaders announced they were withdrawing the country’s ambassadors from France, the US, Nigeria and Togo.
In a statement read out on national television, they said the functions of the four ambassadors had been “terminated”.
Only hours before, Niger’s ambassador to the US, Kiari Liman-Tinguiri, told AFP news agency that the junta “should come to reason” and “realise that this affair cannot succeed”.
Niger is a significant uranium producer – a fuel that is vital for nuclear power – and under Mr Bazoum was a key Western ally in the fight against Islamist militants in West Africa’s Sahel region.
In his newspaper article, Mr Bazoum warned the coup, if it succeeded, would have “devastating consequences for our country, our region and the entire world”.
“Fighting for our shared values, including democratic pluralism and respect for the rule of law, is the only way to make sustainable progress against poverty and terrorism,” Mr Bazoum wrote.
“The Nigerien people will never forget your support at this pivotal moment in our history.”
Mr Bazoum also warned of the coup leaders’ links to Russian mercenary group Wagner, which operates elsewhere in the region and has been seen by many as exercising a malign influence in Niger.
“The entire central Sahel region could fall to Russian influence via the Wagner group, whose brutal terrorism has been on full display in Ukraine,” wrote Mr Bazoum.
Many supporters of the coup in Niger have been chanting pro-Russian slogans and wearing the colours of the Russian flag.
On Thursday, thousands of people took to the streets of Niger’s capital, Niamey, in a peaceful demonstration backing the coup and criticising other West African countries for imposing financial and trade sanctions on Niger.
There is no indication that Wagner was involved in the overthrow of Mr Bazoum, according to the US – but Wagner’s leader has reportedly described the coup as a triumph. The Russian government, however, has called for the ousted president to be returned to power.
The military takeover has also been internationally condemned, including by the EU, UN and the US.
Earlier this week, US Secretary of State Antony Blinken spoke to Mr Bazoum on the phone, with the US saying afterwards it was committed to the restoration of Niger’s democratically elected government.
Mr Bazoum, the first democratically elected president to succeed another in Niger, was detained by his own guards last week. Coup leader Gen Abdourahmane Tchiani has been installed as head of state.
Niger is a key part of the African region known as the Sahel, an area plagued by jihadists and beset by military regimes. In recent years it had been seen as an example of relative stability, while its neighbours Mali and Burkina Faso succumbed to military coups.
It hosts French and US military bases which are used to fight Islamist insurgents.
President Bazoum’s government has been a partner to European countries trying to stop the flow of migrants across the Mediterranean Sea, agreeing to take back hundreds of migrants from detention centres in Libya. He has also cracked down on human traffickers.
