Read Rothbard again - his point is that The Free State An-Caps property rights were not violated. If An-Caps accept the concept of libel, they are accepting the rules of "the State", which they claim to be illegitimate.
You are correct that no property rights were violated. That's not my point. Suing an imaginary entity for a violation of an imaginary rule also does not violate property rights. Not when that imaginary entity can only exist by violating those rights. It's like criticizing a child playing make-believe for following the rules of a made up game.