Mr Rajapaksa, who fled to Singapore this month after he was ousted from office over Sri Lanka's economic collapse, was defence minister during the country’s civil war, which ended in 2009.
The International Truth and Justice Project (ITJP), an evidence-gathering organisation administered by a South Africa-based non-profit foundation, said on Sunday that its lawyers had filed the complaint requesting his immediate arrest.
The complaint alleges Mr Rajapaksa committed grave breaches of the Geneva Conventions during the civil war “and that these are crimes subject to domestic prosecution in Singapore under universal jurisdiction”.
Sri Lanka’s economic crisis has left the nation’s 22 million people struggling with shortages of essentials, including medicine, fuel and food.
Months of protests have focused on the Rajapaksa political dynasty, which has ruled the country for most of the past two decades.
“The economic meltdown has seen the government collapse, but the crisis in Sri Lanka is really linked to structural impunity for serious international crimes going back three decades or more,” ITJP executive director Yasmin Sooka said.
The UN conservatively estimates that the civil war killed 100,000 people but the actual number is believed to be much higher.
A report from a UN panel of experts said at least 40,000 ethnic minority Tamil civilians were killed in the final months of the fighting alone.
Tamil Tiger rebels fought to create an independent state for ethnic minority Tamils.
The country’s ethnic Sinhala majority credited Mr Rajapaksa and his elder brother Mahinda Rajapaksa with the war victory, cementing the family’s political dominance.
Efforts to investigate allegations of war crimes were largely suppressed under the Rajapaksas.
After Gotabaya Rajapaksa fled the country earlier this month, politicians elected Ranil Wickremesinghe to serve the remainder of his presidential term.
He declared a state of emergency with broad powers to act to ensure law and order, and a day after he was sworn in, hundreds of armed troops raided a protest camp outside the president’s office, attacking demonstrators with batons.
Rights groups have urged Mr Wickremesinghe to immediately order troops and police to cease using force and said last week's display seemed to follow a pattern of Sri Lankan authorities forcefully responding to dissent.
The political turmoil has threatened Sri Lanka’s potential for economic recovery.
Mr Wickremesinghe recently said bailout talks with the IMF were nearing a conclusion.