Is it ok to keep lettuce in water in the fridge?

I don't see any reason for it to be unsafe.

However, I think you are wrong in assuming it will keep longer. I keep my lettuce in a closed container with water on the bottom, then a trivet, then the lettuce on the trivet, raised above the trivet, and the whole thing in the fridge. It keeps that way for weeks, not getting yellow, and not wilting too much.

If a leaf falls into the water, or the water is so high that the bottom of the lettuce touches the water, these parts rot within 2-3 days, not unlike how the leaves of flowers rot soon if submerged in the vase while the leaves and blossoms above the water level keep well. So, I don't see why you would place the leaves in water on purpose. Since the rot is easily perceivable, I would say it's safe (you won't eat it once it turns bad), but it is not really a good storage strategy.

The way mold grows is, a spore has to land somewhere moist, i.e. with at least a tiny amount of liquid water on it. So if you want to keep stuff like fresh berries in the fridge for many days, the best thing to do is not wash them right away, but instead leave them with a nice dry outer surface. That way, even though they're obviously full of wet juice inside, they'll last longer because if a spore lands on the outside, it doesn't have access to enough water to grow.

That said, you're typically storing lettuce in the fridge for a few days at most, and one of the other ways to keep stuff fresh in the fridge is by having it completely submerged in water. If a mold spore lands on the surface of clean water it's also not likely to prosper, because it needs nutrients as well as water. This is why if you have sauer kraut or something, you should make sure to push all the vegetables down into the brine (I mean, the salt obviously helps quite a lot there but it's the same basic principle.)

Basically mold grows fastest where there is moisture/condensation on the surface of food - either drier or more water-logged conditions can slow it down.

I've never actually stored lettuce the way you're describing, but I have stored fresh herbs that way and it works great. Herbs completely submerged in water last a while (over a week) and seem perfectly safe, so why not lettuce?

The safety of storing your lettuce submerged would depend on what type of bacteria are on it before storage.

Anything from botulism to E coli could be hanging out in your wet salad.

Both of these, and other, bacteria can survive refrigeration, submersion. E Coli can apparently survive highly acidic environments and fermentation.

That said, it's the presence of the bacteria itself that causes these issues. Good washing can help mitigate the risk. Submersion does give you the added benefit of preventing mold, and mold can be pretty scary too.

I have been keeping lettuce in the refrigerator with only the stems submerged in water for up to three weeks, taking off leaves as needed.