Article VI's Supremacy Clause explicitly binds state judges to the Constitution, federal laws, and treaties as the supreme law of the land, "any Thing in the Constitution or Laws of any State to the Contrary notwithstanding."
The clause creates an imperative duty: when a state judge reviews a case controlled by federal law, they must grant the relief federal law requires, even when state law conflicts. This isn't merely a personal obligation but a requirement of their judicial office.
The Supreme Court has consistently held that when state and federal law clash, federal law prevails. State judges can't cite their own constitution or statutes to avoid federal mandates. The provision was drafted to prevent the chaos of conflicting state court rulings that plagued the Articles of Confederation, where state courts often ignored national laws.