Two Dutch citizens were among 30 activists on board the Arctic Sunrise, the Dutch-registered Greenpeace ship, which was seized by Russian authorities last month after activists staged the protest at the Prirazlomnaya platform.
Russian authorities have pressed piracy charges, which could result in prison sentences of 15 years, against the activists.
The Russian government declined to make an immediate comment.
"The Netherlands today began an arbitration procedure on the basis of the (United Nations) Convention on the Law of the Sea," Foreign Minister Frans Timmermans wrote in a letter to the Dutch parliament.
The Dutch government could "ask the Tribunal on the Law of the Sea for temporary measures for the release of the ship and its passengers", if the arbitration does not result in their release, he said.
The Hamburg-based International Tribunal for the Law of the Sea was set up in 1996 to help settle maritime legal disputes between states.