I have heard that it will "wash away the nutrients", that it dilutes stomach acid and all kinds of things. Is this just an old-wives tale?
This positively reeks of old-wives-tale and pseudo-science. It appears to be widespread online, mainly in 'e-health' forums. It makes sense - water dilutes acid, right?
Right. But diluting an acid doesn't make it weaker. It may slow the rate of reaction, but gastric juices are strong stuff: a glass of water will hardly touch them. It takes an alkali or amphoteric like sodium bicarbonate to neutralise them effectively. And it so happens that sodium bicarbonate is exactly what the body secretes to neutralise the gastric juices. It's also what antacid tablets are made of.
See this answer from the Mayo Clinic:
There's no concern that water will dilute the digestive juices or interfere with digestion. In fact, drinking water during or after a meal can actually improve digestion. Water and other liquids help break down the food in your stomach and keep your digestive system on track.
I'd rather believe a qualified gastroenterologist than a web-full of random forum posts.
Gastric acid production is carefully regulated. Therefore, drinking water (in reasonable quantities) is very unlikely to have a significant negative impact. (After all, any food you eat will also dilute your stomach acid, and the food very likely has pH buffers in it, unlike water.)
Also, "wash away the nutrients" is particularly silly--wash it away to where? It's all still in your stomach, and you have twenty feet of intestine devoted to extracting nutrients from material in your digestive tract.