South Korean President Lee Myung-bak said South Korea will "thoroughly retaliate against North Korea" if provoked again," a report by Yonhap news agency said.
Last week Lee and Prime Minister Kim Hwang-sik visited the National Cemetery in Daejeon to pay respects to the dead sailors.
The 1,200-ton naval corvette Cheonan sank rapidly after an explosion from a suspected torpedo ripped the vessel in half in March 2010. It sank just more than 1 mile southwest of Baeknyeong Island near the de facto sea border with North Korea.
North Korea consistently denies it had anything to do with the sinking.
Tensions rose further that year after North Korea unexpectedly shelled the South Koran Island of Yeonpyeong, killing two marines and two civilians, as well as damaging houses and military buildings.
The latest warning by Lee comes after another South Korean warning over next month's planned test launch by North Korea of a long-range rocket with a satellite that Pyongyang said is for peaceful Earth observation work.
North Korea said it will launch the satellite between April 12-16 to mark the 100th birthday of the state's founder Kim Il Sung on April 15.
South Korea said it would consider shooting down any North Korean rocket that strays into its territory, a report by the BBC said.
Yoon Won-shik, a spokesman at South Korea's Defense Ministry, said the government is "studying measures such as tracking and shooting down (parts) of a North Korean missile in case they stray out of their normal trajectory" and violate South Korean territory.
Japan also said it is monitoring North Korea's plans for the launch and would consider shooting down the rocket if it strayed into Japanese territory.
The order to the Self-Defense Forces could be given after a meeting of the Security Council of Japan led by Prime Minister Yoshihiko Noda, the Yomiuri Shimbun reported Monday.
Japan will deploy three destroyers in the East China Sea, the Pacific Ocean and the Sea of Japan to track the missile's path, military sources told the newspaper. Also, the ministry said it will deploy ground-based Patriot Advanced Capability-3 interceptors.
Despite regional concerns over North Korea's launch, this week Seoul also is the venue for governments, including the United States and China and anti-terrorist organizations attending a nuclear security summit that began Monday.
Representatives from more than 50 countries, including U.S. President Barack Obama, are in Seoul for the second round of meetings aimed at keeping nuclear materials out of the hands of terrorists.
The first summit was in Washington two years ago.
Curing a meeting on the sidelines of the summit, Chinese President Hu Jintao told Lee that "at present, the situation on the Korean Peninsula is very complicated and sensitive," are report by China's state-run news agency Xinhua said.
Hu said China -- the main ally of North Korea -- will work with all parties for peaceful resolution of conflicts and tensions, Xinhua reported.
During a meeting with Obama, Hu said he would work closely with the United States over responses to concerns about North Korea's planned launch.
"The two leaders agreed to coordinate closely in responding to this potential provocation and registering our serious concern to the North Koreans and, if necessary, consider what steps need to be taken following a potential satellite launch," said Ben Rhodes, a senior White House aide.
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