Unfortunately, the best fabric for water slings (solarweave) is no longer in production - there are a few carrier makers who have small stocks of it but it's hard to find, and not cheap. The breeze baby ring slings are a good option. The big problem with many mesh type fabrics (especially ones that can be bought retail) is that they are not strong enough for babywearing, especially if you are going to be using it in chlorine or salt water. Something like solarweave is breathable but a strong weave and synthetic thread, so it stands up to being used in the water.

You could use linen (if using handkerchief weight use two layers), but it's going to feel a bit yucky against the skin when wet, will be difficult to adjust when wet and won't dry as quickly, and you'll have the same problem with fabric degradation if using it over the long term.

If you specifically want a ring sling (something that has seams and stress / adjustment points) I would invest a little bit more and get something that can be used safely and comfortably, especially as your LO gets a bit bigger. For small babies, a cotton gauze wrap could be made quite cheaply (think sarong weight type fabrics), but with any fabric used in water be careful to check it before each use for degradation.

HTH!