Two completely different things.
The ICC, operating under the Rome Statute, deals with prosecutions of crimes committed by individuals -- war crimes, the crime of genocide, crimes against humanity. (The US and Israel have both signed but not ratified the Rome Statute.)
The ICJ ("World Court"), operating under the United Nations Charter and the Statute of the ICJ, adjudicates claims by one state against another relating to breaches of the international law that applies between states.
One doesn't "file a lawsuit" in the ICC. States refer situations to the prosecutor and request investigation, after which there may be prosecutions of an individual or individuals.
Conversely, one doesn't prosecute in the ICJ, which is essentially a civil court. It doesn't try or punish individuals; it decides disputes between states and can award compensation.
It looks like the Lebanese minister is asking that both things be done:
"... a bid to punish these crimes and to bring them to justice ..."
"... with the intent that Israel pay restitution on all the physical and moral damages that she caused Lebanon and her citizens ..."
"... to assign the mission of performing a comprehensive survey of damage caused *and* of gathering proof of crimes committed ..."
-- which seems wise.
Just another note in case of confusion: "moral damages", referred to in the article in the opening post, is an expression used in European civil law for what we might call "non-material damages", i.e. harm that can't be measured in dollars and cents but for which someone is entitled to compensation.
And I'd love to see an Iraqi government go after the US sometime in future, too, in particular in the ICJ for violations of international law, i.e. the illegal invasion and occupation.
Of course, the US withdrew from compulsory jurisdiction of the ICJ in 1986. The US lost in the case brought against it by Nicaragua for the mining of its harbours by the US. Withdrawing from compulsory jurisdiction is an appalling affront to the world, and destabilizing and decivilizing influence on international relations. Declarations by other nations regarding compulsory jurisdiction:
http://www.icj-cij.org/icjwww/ibasicdocuments/ibasictext/ibasicstatute.htm Website of the ICJ:
http://www.icj-cij.org