Can a Washington State Judge rule on a business owned and operated in South Africa

If the business has a sufficent number of ties to the U.S., such as if it sells or advertises products or services in the U.S.--and more, in the county in which the suit has been brought--then the court may have jurisidcation over the business. If it does, it has the legal authority to issue an order or judgment binding on that business, though enforcing it may be different matter (see below).
Similarly, if you have sufficient ties to the U.S.--e.g. you live here--and again, specifically in the county in which the court is located, the court may have jurisdiction over you personally; and if so, it can issue an order that you pay some portion of the money you receive from the business--though again, enforcing that order may be a different story.
If neither you nor the business have sufficient local ties, the court may have no jurisdiction or power over you; so if you and the business are in South Africa and do not do business here or travel here, it is difficult to see that the court would have power over you. In this circumstance, the case may be dismissed for lack of jurisdiction.
Even if the court has jurisdiction over you and/or the business, if all the assets are in South Africa, even after getting a judgment against you, the plaintiff (person suing) would have to then try to get a South African court to enforce it against you, since a U.S. court has no actual abilty to, say, garnish or levy or execute upon income and assets in South Africa. While getting a South African court to enforce a U.S. judgment is possible, it is a multi-step, somewhat complicated, and time-consumming process.