Willingly, yes. Unwillingly, probably.

The Ring of Water Walking has the following rules text:

While wearing this ring, you can stand on and move across any liquid surface as if it were solid ground.

Being willingly pulled under or diving

First I note the word can which makes it clear that the wearer has to choose to use this effect. Since there is no defined timing restriction on when they can make that choice I will assume they can do so at any time. I also note that this effect only applies to a liquid's surface, but doesn't explicitly do anything when you are submerged.

Thus you can always end the effect of the ring and let yourself be pulled underwater or simply dive.

Being unwillingly pulled under

This ring allows us to treat the surface of water as though it were solid ground, but just like solid ground it is by nature an uneven surface under which creatures dwell.

If a burrowing creature could reasonably grapple a target and pull it underground using its burrow speed, I see no reason why a swimming creature couldn't grapple a target and pull it underwater using its swimming speed. Just like being pulled underground you're not on a solid surface anymore as it is being parted to pull you in.

As soon as you start sinking the Ring of Water Walking will no longer be of any use to you given that you are not on the water's surface anymore.

Just like being pulled underground you could try to brace yourself onto the surface with your arms but allowing for this is squarely in DM territory.