Interstate commerce belongs to congress (Not the president!). Congress has allowed California to ban registration of cars that do not meet their emissions standards (other states are allows to take California's standards, but are not allowed to make their own). California can constitutionally create their own standards for cars made or sold in California, but because they are part of the US they have to register a car you bought in any other state which means their standards hurt local dealers.

Note that there has always been an exception to the rules: if you are out of state and your car it totaled you can buy a new non-California car and register it in California.