Unless its owned by the Army Corps, I don't think they have any say in the matter. It's up to a court. To even make it to court, it has to be challenged, meaning someone has to cross the posted signs claiming navigability, and the landowner has to attempt to prosecute, and both sides have to hold their ground the whole way through a lengthy court battle.
Officially, if the stream is capable or has been capable for "commerce", in its natural condition, then its navigable. A judge will look for a history of "commerce", which is not simply running a kayak down it. A legislative declaration of public highway is looked highly upon, but I don't think that exists for the Pickering (at least I couldn't find one). I'm unsure how an inner tube business would be viewed. But if you could come up with historical evidence that Pickering was used to transport something or other for business purposes, and that at least one of these things did not require dredging, canals, or splash dams, then you may have a case.