Fair Courts Act of 2023 This bill prohibits a federal court from granting a form of relief that has nationwide applicability unless the case is heard and determined by a panel of three district court judges. This relief includes granting an injunction that restrains the federal government or federal officers, an order that vacates a federal regulation, or a stay of any such relief. The bill also requires a district court to ensure that a case does not have greater than a 25% chance to be assigned to particular judge. The chief judge of the circuit is allowed to assign judges from adjacent districts to other districts that do not have enough judges to comply. District courts must also publish orders regarding the division of business among judges.