BRUSSELS—Just weeks after joining the European Union, Croatia is embroiled in a clash with Brussels that could see it lose access to some funds and kept out of Europe's border-free Schengen accord.

The European Commission, the EU's executive arm, has given Croatia until midnight Friday to respond to a letter from Justice Commissioner Viviane Reding demanding the Croatian government scrap an amendment to its domestic law implementing the European Arrest Warrant, which obliges member states to extradite citizens to another EU...