South Koreans have been staging largely peaceful anti-Japan rallies near the Japanese Embassy in Seoul for decades.
South Korea and Japan, both key U.S. allies, have been often embroiled in history and territory disputes stemming from Japan's 1910-45 colonial rule of the Korean Peninsula.
Heightening the tensions is Tokyo's recent decision to restrict exports of high-tech materials to Korea, which Korea believes is a retaliatory measure in protest of the Supreme Court's rulings.
South Korean officials have said that Japan's call for an arbitration panel is unacceptable.
On Friday, Kono summoned the South Korean Ambassador to Japan saying it was "deeply regrettable" that the deadline had passed.
An elderly South Korean man died on Friday after setting himself on fire outside the Japanese embassy in Seoul as a bitter diplomatic dispute over wartime forced labour compensation took a fatal turn.More news: Australia 'deeply disappointed' by writer's detention in China
Early this month, Japan announced restrictions on the export of several chemicals to South Korea that are key to the chip and smartphone industries.
The row began after South Korea's Supreme Court ordered Japanese firms past year to compensate forced labor victims.
Japan says the issue of compensation was settled under a 1965 treaty which established diplomatic relations between the two nations post World War Two.
Tokyo had asked South Korea to hold an arbitration meeting by Thursday on the issue.
The man, who was in his 70s, started the fire inside his auto as it was parked in front of the building where the Japanese Embassy is located, according to police in Seoul.
Kono claimed that what South Korea is doing amounts to "overturning the global order since World War II" and demanded Seoul take steps to correct the situation.
Later, South Korea's foreign ministry rejected Japan's arbitration call as arbitrary and said Japan must remember its wrongs committed during colonial rule and try to heal the wound.More news: Trump, the 'Squad' and the 'send her back' chants
Japan has said it implemented the export curbs over South Korea's "deficiencies" in its export control systems and not in reaction to the labourers' dispute, but South Korea has called the restrictions "unjust economic retaliation".
South Korea could not ignore its Supreme Court ruling ordering compensation, he said.
The man's father-in-law was said to have been a victim of forced labour during World War Two and the man may have acted in protest against Japan's export curbs, media said, citing police.
The curb on exports of specialty materials vital to South Korea's technology sector triggered concerns on disruptions in the global supply chains and declines in tech stocks.
South Korea's Samsung Electronics has sent letters to partners urging them to stockpile more Japanese components in case Tokyo expands its export restrictions. Japan denies that, saying the controls are required for national security.
Japanese Prime Minister Shinzo Abe greets South Korean President Moon Jae-in during the Group of 20 summit in Osaka, June 28.More news: Neymar: Barcelona up the money stakes for PSG star
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