If the EU had direct legislative powers, it would most likely be structured by letting it legislate EU law, not the national laws of the member states, the same was US federal law is interpreted under US federal law, not under the laws of individual states.

We already have experience with this, in that the ECJ is already the highest court for matters of European Union legal matters, such as the interpretation of directives, and national courts in EU member states will often need to interpret cases in light of EU law.