Arriving in South Korea Wednesday from China, Clinton issued her appeal as relations between the two Koreas plunged further with the North severing all ties with the South.
An international team of investigators last week said North Korea was responsible for the March 26 sinking of the South Korean war ship Cheonan in the Yellow Sea, killing 46 sailors. The North has denied any involvement.
"This was an unacceptable provocation by North Korea and the international community has a responsibility and a duty to respond," Clinton told reporters in Seoul.
Separately, South Korea Wednesday vowed to press on with retaliatory actions against the North, Yonhap news agency reported quoting Unification Ministry spokesman Chun Hae-sung.
"Even though the North should apologize and prosecute those responsible, it has again taken measures undermining inter-Korean relations," Chun said. "The South will deal with these North Korean threats unwaveringly and sternly."
Prior to Clinton's arrival, South Korean President Lee Myung-bak announced a number of measures against the North including the suspension of all trade.
Yonhap reported in Beijing, Clinton tried unsuccessfully to convince Chinese leaders to forge a united, tough response against the North.