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This wording misstates what happened at the International Court of Justice.

The UN General Assembly did not "take the world's biggest polluters" to the ICJ as parties in a case. Instead, it requested an advisory opinion from the Court on the general obligations of states in respect of climate change. The ICJ's own case page says the General Assembly asked the Court to give an advisory opinion on two legal questions: what obligations states have under international law to protect the climate system, and what legal consequences follow when states have caused significant climate harm.

That matters because advisory proceedings are fundamentally different from suing particular countries. The ICJ explains that, in advisory jurisdiction, public international organizations cannot be parties to a case, and that this special procedure is used to obtain the Court's legal opinion. The ICJ also says advisory opinions are generally not binding. So this was not the UN hauling identified "biggest polluters" into court as defendants; it was the UN asking the Court for legal guidance on climate obligations.

A more accurate description would be that the movement helped spur the UN to seek an ICJ advisory opinion on states' climate obligations, rather than to "take the world's biggest polluters" to court.