And I am saying this as someone in the middle of moving to New Orleans, a place humanity has been fighting with a river over for three centuries now. Someday the Mississippi will assert its claim forcefully. And New Orleanians will probably just shrug and either build up houses that sit next to canals instead of streets, or learn to grow gills. New Venice or New Atlantis.
It'd be a nice change from activist judges who take the side of the corporations.
People have sued God (and lost): https://en.wikipedia.org/wiki/Lawsuits_against_God
People have sued search warrants (and won): https://en.wikipedia.org/wiki/Marcus_v._Search_Warrant
So you probably can! But, lacking a fixed address or any way to notify the defendant, you'd probably have the case thrown out. Also, enforcing any judgement would probably be impossible, so there's that.
The moral of the story is that courts have lots of ways to deal with situations like this.
Compare with what personhood has done for corporations - classing political donations as "free speech" has been an avenue for them to get around limits intended to make it harder to buy politicians.
We speak metaphorically about how corporations rape the land. Can this river now sue corporations who dump garbage into it for raping it? Probably not if there's a legal definition of what "rape" is. But maybe charges of "assault" are possible.
The article notes that which rights a body of water has have yet to be determined. That's going to be some interesting times.