Be careful with using river water to accomplish this. River water is often full of bacteria and contaminants that can cause cuts to become infected or slow healing. A better choice would be to boil the water and add about a teaspoon of salt to one quart of water and use that to flush the wound.
An interesting suggestion. I would be careful to make sure the cut is minor. I wouldn't recommend this for more serious lacerations.
I'd also recommend to repeat it as needed, and apply a balm such as bag balm, or a medicated balm such as neosporin or similar brands. Though I honestly prefer bag balm. Try not to use too much, only a little.
Also I am unaware of the effect of salt for cuts. River water seems nice coupled with thoughts on a speedy recovery. Then a light application of balm.
I do feel the wording it will soon go away may need addressing. Speeding healing with methods such as you suggest, may lessen pain, or speed up recovery a little. But this tends to take days or weeks, and will not suddenly go away, but will appear to grow shorter, and grow scabbing over time, for a minor cut.
Also if the cut is not on the leg and in an area difficult for this method, chest; under the arm; it may prove a bit more difficult to use the method in river water.
All in all the process used here can be sound, but is not the only method, and may not entirely be necessary. As well, it will take time to heal. Even if this suggestion speeds up recovery, and this as well can occur naturally.
The effectiveness of salt comes down to the function of osmosis; the natural transfer of water from areas with low density of materials towards areas with high density of materials to create an equilibrium.
Salted water has higher content than the water in the cells exposed to it, so water comes out of the cell to dilute the water outside and equalize the ratio. Sort of the reverse of air pressure if you look at it the right way.
This also means that it dessicates bacteria (dries them out) and kills it, often also drawing the material/pus out of an injury. This is a very useful treatment for surface infections like those from hang-nails and other skin sores on the hands or feet. And if you have ever o served the wound after it has been soaking for 30-minutes or so, you might see a stringy bit floating in the water from the infected site. This is said pus, having been (slowly) pulled out.
Everyone begins somewhere stover, try to not take our words as admonishment, but as education. You were right that salt has cleansing properties which, when applied correctly, does improve healing. And it is also true that while not perfect, moving water is the safest to interact with as still or stagnant water is rarely free of mud or contaminants. This is why if you are in a survival situation you avoid drinking from puddles or still water. It can be almost worse for you than going thirsty.