"These violations of the genocide convention are a stain on our collective conscience and it would be irresponsible to pretend that it is not our business," the Gambian lawyer told judges.
In court on Monday, Myanmar struck out at The Gambia for having brought the case before the Hague-based ICJ, set up after World War Two to rule in disputes between countries.
Its lawyers accused Banjul of not acting as a "country in its own right", but as a proxy for the Organisation of Islamic Cooperation, a 57-member body set up in 1969 to represent global Muslim interests.
Jallow dismissed that argument.
"This is very much a dispute between The Gambia and Myanmar," he said.
"We make it our business when we, as civilised nations committed ourselves to the pact under the 1948 Genocide Convention," Jallow added.
When the case opened in December 2019, Aung San Suu Kyi personally represented Myanmar at the ICJ, but she was ousted as the country's civilian leader in a military coup last year.
The Nobel peace laureate, who faced criticism from rights groups for her involvement in the case, is now under house arrest and on trial by the same generals she defended in The Hague.
Around 850,000 Rohingya are languishing in camps in neighbouring Bangladesh while another 600,000 Rohingya remain in Myanmar's southwestern Rakhine state.
The Gambia's lawyers, quoting recent human rights organisation reports, said the Rohingya remained vulnerable.
"The Rohingya remain at grave risk of mass atrocity crimes," one lawyer Paul S Reichler warned the court.
The Rohingya case at the ICJ has been complicated by the coup that ousted Suu Kyi and her civilian government, triggering mass protests and a bloody military crackdown.
More than 1,500 civilians have been killed, according to a local monitoring group.
The ICJ made a provisional order in January 2020 that Myanmar must take "all measures" to prevent the alleged genocide of the Rohingya while the years-long proceedings are under way.
But the European Union on Monday heaped more sanctions on Myanmar officials, saying it was "deeply concerned by the continuing escalation of violence in Myanmar and the evolution towards a protracted conflict with regional implications."
The bloc added 22 junta officials, taking a total of 65 on the sanctions list, and four companies tied to the regime, making 10 overall.
Among those targeted were the ministers for investment, industry and information, officials at the election commission and senior members of the military.
The ICJ judges must now decide whether the court has the jurisdiction to hear the case. If so the case could still take several years.
Decisions by its judges are final and cannot be appealed, although the court has limited means of enforcing them.