Eroding River Rock

Not likely to be a good solution...concrete must have a proper base and thickness to support anticipated loads...why not just add more rock?

Your question makes no sense. "River" rock is just rock.

You don't say what size?

What kind of rock? (Other than round)

"River" rock is round from being rolled downstream for millions of years and does not normally have built up internal stresses which would cause it to crack without outside stress.

Finally, one inch of concrete over rock will crack.

Pictures or more information will help.

On shoreline they would put it in riprap, to protect against erosion.... Of its not very scenic.

Perhaps a binder to make the rock pebbles into larger block may help a little to hold on for few decades, but its top surface going to experience erosion from run over by smaller rolling rocks all the time. If water flow is regulated then it may work much better to reduce the erosion rate. Some kind of dam may be built and then water can be channeled out in a controlled manner for irrigation etc. Whatever is planned may work only for few decades and not for millions of years.

I have seen seashore temples of rock being eroded just by wind in over 1000 years. This seashore temple was built at Mahabalipuram, TN, India by Pallava dynasty in sand stone rock is now badly eroded.

Assuming this is a horizontal surface that you are referring to, you might consider an exterior grade of concrete leveller such as Levelfloor tuff or similar.

They have good durability from paper thin up to about 1" or even thicker if you add fines.

Make sure you clean and prime the surface if you go this way.

Elastomers can often be added to improve flexibility.

Exactly--There are epoxy modified coatings , that will cover , provided the substrate is fixed, and not flexible.. You can get self leveling , highly elastic coatings, for specific projects also.. Is the area compromised by underslab moisture intrusion? IE..Do you have water running at times under the rock? Does the area lift in hydrostatic conditions? Sink in dry conditions? If so, surface coatings will, in the long run, be of no use..Build for the conditions you have. Get a soil moisture meter, and run it up and down the area to be covered. It will tell you many things...

How is that river rock is breaking down. Those round rocks hardly ever crack. Or are you saying tiles made with round stone?

You can delay the cracking of the rock by applying a silicone based UV protectant & water proofing product such as "Thompson's Water Seal".

If the issue is with the grout lines you will have to remove and replace the bad areas then coat them with the same protectant product.

The coating will have to be refreshed/re-applied on an interval that meets OEM specifications.

If you have to replace the grout there are "plastic-based" grout products available that remain flexible which will minimize and/or stop cracking.

Keep in mind that the rock and grout expand and contract at different rates so you must have expansion joints placed at key intervals of no more than 10 feet distance.

Hope this helps.

I don't know if you can get this in the US.

Heidelberg "FlowCrete" self leveling screed. It was being launched in Europe about 15 years back just as I left the company.

It will flow between the "river rock" (whatever that is) and bind it.