"North Korea must admit its wrongdoing, it must pledge to never again engage in such reprehensible action.
"This is in the interest of peace. This is in the interest of North Korea."
Last month, a team of international investigators concluded that the Cheonan was hit by a North Korean torpedo.
Pyongyang has denied having anything to do with the incident and has accused the South of fabricating evidence and framing it for the attack.
But Lee called the March 26 sinking, in which 46 South Korean sailors were killed, a "military provocation" and he dismissed the North's denials of involvement as "laughable".
However, he did not mention sanctions, and it was not immediately clear what action Seoul wanted from the Security Council.
"If the enemy continues to taunt us and think that they can do whatever they want, they must understand that there is a limit ... that they have to suffer the consequences," he said.
The spat has already seen the North cut all ties with the South, scrap pacts aimed at averting accidental flare-ups along their disputed sea border and vow to attack any intruding ships.
The row has also caused a diplomatic headache for China, which maintains good relations with both countries.
Lee has indicated that he will continue to pressure Beijing to support a Security Council resolution against the North.
Meanwhile, the US has announced that its forces will conduct joint naval exercises with South Korea "to ensure readiness and to deter future aggression" by North Korea.
But Robert Gates, the US defence secretary, who was also attending the forum on Friday, said that a joint show of force by the US and South Korea may be put off to allow time for the UN to consider the allegations against the North.
"I think there's a sequencing involved in this and it may be there's a desire first to see what can be accomplished at the UN and then think about next steps beyond that," he said.