At the risk of oversimplifying this - the answer to your question is no. There are 4 navigability standards which each have different consequences. You are referring to navigability for title purposes. If a river is navigable for title - which means, more or less, that the river could have supported commerce about a hundred years ago - then the land up to the normal high water line is held in the public trust. So, even if you own the land on each side of the river, title to the river bed is owned in public trust by people of the United States collectively. (This standard is not really clear, and there are weird applications of it all over. Apparently, there are no navigable waters in the entire state of Colorado under this test.)
BUT... just because you are part of the group that owns the land doesn't mean you can do whatever you want on it. Think of it like public lands - forest service, BLM, national parks etc. Each are technically owned by the people of the US, but just because you are a citizen doesn't mean you have free reign to do whatever you want in forrest service land. The government decides the best use of the land and imposes the rules that everyone must follow. So just as the government can decide whether or not you can use a dirt bike on a forest service trail, they can also decide to implement river permits for the preservation of certain rivers.
The navigability rules only stop a private land owner from claiming trespass, they don't stop the Gov't from regulating the river in anyway they want.
That said, I'm not sure if anyone has ever tried this argument. So you can try to convince a judge to listen to you - it might work. Crazier legal decisions happen all the time. Also a lot of permitted/illegal rivers get poached every year. Remember, you only have to make the argument if you get caught.