Washington says the US navy attacked the platforms because they were being used for attacks on shipping during the Iran-Iraq war.
The US has filed its own case, calling for Iran to pay damages for threatening shipping in the Gulf.
The hearing of testimony is scheduled to last three weeks but a judgement may take months or even years to be reached.
Iran filed its complaint to the court in 1992, saying that the US attack on its oil installations violated a 1955 treaty of political and economic friendship between the two countries.
US warships destroyed the first platform in October 1987 after an Iranian missile injured 18 crew on a US oil tanker.
Washington - which supported Iraq in the war - ordered two more off-shore installations to be destroyed in April 1988 after 10 sailors on a US frigate were injured in a mine explosion.
The US said the oil platforms were being used to launch attacks on shipping in the Gulf, but Iran denied the charge.
Binding ruling
The US objected to the filing of the case in 1992, arguing that the ICJ did not have jurisdiction.
But the court - which sits in The Hague - rejected that in 1996.
Judges did, however, allow Washington to file a counter-claim seeking damages from Iran for also violating the 1955 treaty and endangering commercial shipping.
While the court's judgement will be final and binding, it has no means to enforce any ruling.