Well, it's just a compensation or balancing system. In Aperture priority you turn one and the camera turns the other - In Shutter priority you turn the other and the camera turns the one. It's a compensation system, a balancing act if you will where the two ends of the rope are A and S and the general height of the rope is the ISO.
Remember, YOU are the driver. So you set it to the priority YOU want. If YOU want control over the shutter speed (for effects like this) then give yourself Shutter priority. For affects like this you need the shutter to be at least 1/60. 1/30 or 1/20 makes the affect even more dramatic and 1 second will make it look like a river of ice and milk (kinda).
So set the shutter to 1/60 or 1/50 and let the camera set the aperture. As you adjust the Shutter (with your finger on the trigger) if the camera pegs the aperture before you reach 1/60 then you need to lower the ISO. With many if not most lenses in many if not most daylight conditions for many if not most river scenes you should have enough room to set the shutter to 1/60 or so at ISO 100 (or lower if you have it), and have the camera wanting to set the aperture somewhere between 8 and 22. No need for an ND filter at that point.
If you still can't get there however there is one more control available on most cameras: "Exposure Compensation", AKA: "Exposure Value", AKA: "EV". Different cameras will have different min./max. values but usually it's -3/+3. So start cranking that down as well. Different DCs handle EV differently so not in all systems will this help and it may not be allowed or it may defeat your purpose. It's digital film so nothing is wasted by trying.
If you're at 100 or the lowest ISO and your aperture is pegged wide shut before you reach 1/60 then you might need a filter. Polarizing filters if you have one, are often dark enough and can sometimes work. Additionally they cut glare (specular reflection at particular angles) so it's like turning down the brightness of just the highlights (in a way) and will assist doubly for this purpose in that regard.