After the top Polish court overruled the ECJ, Fears Rose the Court Ruling Points to EU Exit.

Tens of thousands of protesters marched through Warsaw and other Polish cities late Sunday to oppose a court ruling that European Union legal judgments have become incompatible with the Polish constitution, a decision protesters fear could prompt Poland to follow the U.K. out of the bloc.

Waving EU and Polish flags, demonstrators held banners reading "I'm Staying in Europe" and "No Polexit!"

Unlike in the U.K., an overwhelming majority of Poles wish to stay members of the EU—as do Hungarians, another Central European country whose government is in regular conflict with the bloc over where the EU's powers end and national sovereignty begins.

On Thursday, Poland's Constitutional Tribunal ruled that the process of European integration encoded in EU treaty law has reached what it called a "new stage" that is incompatible with the Polish constitution, and that the latter should take precedence when the two conflict. When joining the EU in 2004, Poland agreed to implement EU treaties, also signing up a few years later to the bloc's updated Lisbon Treaty. Poland's ruling party says the EU has overstepped its authority.

In Brussels, a spokesman for the European Commission, the EU's executive arm, on Monday gave no timeline for responding to Poland. EU officials fear a domino effect and gradual disintegration of the EU's legal and political authority if one country can overrule EU rules and EU court decisions.

"If you allow all these fundamental principles of European integration to be hollowed out and ignored, then this is eventually the end of the EU," said Piotr Buras, head of the European Council on Foreign Relations' Warsaw office.