The short answer depends on what "wet" means
If wet means "covered or soaked with a liquid," water is not wet — it is what makes other things wet. If wet means "liquid, or behaving the way liquid does," then water is the wettest thing there is. Both readings are in the dictionary, which is exactly why this argument has survived every attempt to end it. There is no fact being disputed here. There is a definition, and two groups of people reading it differently.
That makes it the purest kind of debate this site hosts: nobody can look the answer up, because the answer depends on which half of the definition you consider primary.
The case for "no"
The "not wet" side leans on how the word is used in practice. Wet is a state that something enters and leaves: a towel gets wet, a sidewalk dries. It describes a surface with liquid on it. On that reading, wetness is a relationship between two things — a liquid and whatever it touches — and a substance cannot be in a relationship with itself. Fire is not burnt; water is not wet.
Physical scientists often land in this camp, with a more technical version of the same idea. Wetting describes how well a liquid spreads across a solid surface, measured by the angle a droplet forms where it meets that surface. Water wets glass very well and wets a waxed car very badly. Under that definition, wetness is a property of the pairing, not of the water.
The case for "yes"
The "wet" side has two strong arguments. The first is everyday language: people say "the paint is still wet" to mean it is still liquid, and nobody objects. Many dictionaries include exactly that sense — consisting of or containing liquid — alongside the "covered in liquid" sense.
The second is physical, and it is surprisingly good. Water molecules stick to each other through hydrogen bonds, which is what gives water its surface tension. Any given molecule in a glass of water is surrounded by, and in contact with, other water molecules. If being touched by water is what makes something wet, then all the water in a glass is touching water, and therefore wet.
Why the internet will never let this go
The question took off online in the 2010s, and it now gets searched an enormous number of times a month — far more than most grammar questions. Part of the appeal is that it feels like it should have a scientific answer and turns out to be a vocabulary question in a lab coat. The other part is that it is unwinnable in the best way: both sides are internally consistent, and neither can be dismissed as uninformed.
So vote on instinct. If you picture a wet dog, you are probably in the "no" camp; if you picture a puddle, you may be in the "yes" camp. The tally will show which mental picture most Americans carry around.
Sources: definitions of "wet" in Merriam-Webster and the Cambridge Dictionary; standard treatments of wetting and contact angle in surface chemistry; basic physical chemistry of hydrogen bonding and surface tension in water.