Real question: is the European Court the successor to The Hague?
I'd bet the UN will try to hang on to whatever judicial jurisdiction or authority they can wield through The Hague International Court. The UN is into trying to expand its influence rather than being replaced by any upstarts.
If by ‘The Hague’ you mean the International Criminal Court, then they aren’t related in any way. The only function of the European Court of Justice is to adjudicate on compliance with EU Treaty law by the member states. Cases are referred to it by the courts of final appeal of the individual member states, or by the EU Commission. The ECJ has no power to enforce its rulings. The only sanctions are fines on non-compliant member states.