Is it possible without magic? Just like our world's rivers, but instead it's a river made out of milk.
Milk is an immensely complex mixture of fats, proteins, and carbohydrates, to provide a complete diet for an infant. These compounds are created in nature solely by living creatures. Even if one posited a single-cell creature that happened to match these proportions when it died and broke down, it would be impossible to concentrate it sufficiently; the environment would be all the creatures, and no food for them, and then the river would sweep them away into more water to dilute them.
Not milk as we know it, no. Mary gave a good definition of milk, and following it the only way to have milk rivers is to have some kind of very strange giant creature that excretes a river of milk its young swim in. Fun idea, maybe not quite fit for every world and would also require some weird evolutionary pressures to evolve.
But if we consider any white-ish liquid that contains fats, proteins and carbohydrates and is basically nutritious to count as milk, we might have some options. I agree that the compounds in milk are pretty much only made in living creatures. But a river could, for example contain (micro)organisms that secrete nutrients into the water, which might be bacteria (as Renan suggests), algae, even plants or animals. It's still a very unlikely adaption and you need to consider that not only would this dilute, it would also go bad in the sun and probably attract all kinds of other lifeforms to feed on it.
Is it possible without magic? Just like our world's rivers, but instead [a] river made out of milk
Not naturally as others have already said.
The only suggestion I can make is that an old civilisation created a vast food-producing factory. When they were wiped out, the factory kept on making its products. Milk, being the only liquid one, eventually spilled out of its tank and started flowing down the hillside. All the solid food containers filled up and became blocked so the factory diverted all its resources to making milk.
For this to continue there must be a huge automated farm somewhere. Instead of cows, there would be genetically engineered "teats" that are constantly supplied with the necessary nutrients from the farm.
Fish have adapted to live in the milk but they have evolved to become blind because it's impossible to see through milk. They don't need vision because all their food comes from the milk itself.
I thought this: a fictional species that is a "hybrid" between amphibian, reptile and mammal that needs to be in water as a cub (like an amphibian), but can live in dry environments as an adult (like a reptile).
After gradual changes in the biome to a desert, those who could still reproduce were those who could create their own small ponds for their young.
The liquid produced would be milk to provide nutrients for them (like a mammal).
Many of those amphibian-reptilian-mammal in one place create a large lake that overflows into a river (depending on the terrain relief).
They could live in groups and then those who produce more milk help those who produce less.
Probably these rivers would be temporary, forming only during the mate season.
But if each group has a mating season at different times, but they happen always in the same place (similar to sea turtles), perhaps the river could be constant (with one group coming after another).
Edit: another advantage of doing this is to attract prey, because they would stay for a long time looking after the young and would not have time to hunt.
Other answers point out that the creation of such a river is highly implausible.
Rivers displace a lot of liquid, it would take a huge energy investment to produce enough milk. What does the organism get in return? It'd have to be a by-product of a huge, localized process. Probably the only energy source is the Sun, so maybe a weird kind of photosynthesis?
But even if it existed, I think it wouldn't last long.
Such a huge amount of easily available nutrition would quickly attract some bacteria or other lifeform that feeds on it. Which would keep multiplying until most of the nutrition is extracted close to the source.
This has been somewhat covered this already, but I wanted to add some thoughts, especially after reading some of the other answers hypothesizing ways that such a river might be plausible after all.
A dairy cow, which has been heavily bred for maximized milk production, produces a peak of about 15 gallons/day of milk (a more realistic number is a bit under half this, but let's be optimistic). If we consider a sizeable herd of 1,000 cows, that's 15,000 gal/day, which sounds like a lot, right? Well... if you do the math, that works out to a bit over 10 gal/min. A "good" faucet (or, let's say, a garden hose) puts out about 2.5 gal/min. A fire hose can be up to 500 gal/min. That means that 1,000 cows can produce a continuous flow of milk equivalent to... 4-10 garden hoses, or 1/50th of a fire hose.
That's not much of a "river" by just about any standard.
The previous answerers do a good job of speculating how this couldn't or could (with caveats) work. I'd just like to add some context with the real-world figures.
According to Statistica.com, US milk production gives about 4,000 liters per second (though I may have made a mistake in calculation -- please tell me if I did!). In contrast, the headwaters of the very small Metolius River in Oregon discharges around 3,000 liters per second (The Ore Bin, March 1972 [data within range]). So, all the cows in the US might produce a small river. But to put all 94.8 million together would take an unimaginable economic and environmental toll on the area around. Heck, their giving the world an unimaginable toll on the world without being crowded together. As @Matthew also calculated, this is just not realistic.
Although animal based milk is probably a no go, how about plant based milk? Coconut milk occurs naturally and plant evolution for items like fruit is based on ways to ensure seed propagation via local consumption and excretion in a far away location by animals. Other plants simply have the seeds stick to the animal to be brushed off later in a far away location.
If the above statements hold, it could be possible for trees to evolve in a way that they produce seed ridden milk they release into water in order to entice some creature to drink the milk and in so doing so either consume their seeds to be excreted later or have the seeds stick to the animal until they get brushed off at a later date. I would imagine this kind of tree would be in a mangrove like environment and its target is lifeforms that live in and around the mangrove.
If this method of plant reproduction is highly successful then the mangrove would continue to grow in size and, what was previously highly diluted milk streams in a river, would eventually turn into a river of milk diluted by water. How far the river would stretch until all the various plants, animals and bacteria consumed the milk is a questionable proposition but it should at least be visible as a river of milk if you are near the source. If people don't think that a continuous river of milk is possible owing to the volume of water, have all the trees "bloom" (deposit their milk and seeds) at the same time, so you get a river of milk for a few weeks per year.
Better propagation is a plus for evolution, so is it is quite possible that the milk evolves an anti-bacterial property like that of honey or egg white which would prevent bacteria from consuming it,thus allowing it to travel further in the river and attract more animals to carry the seeds elsewhere.
In some ways, we are replacing aspects of fruit and wind based seed distribution with water based distribution, so the trees being an area with little wind, few flying insects/birds and lots of water makes sense.
One caveat is that this is technically not a river of milk. It's a regular water based river upstream from the mangrove that becomes heavily poluted with milk in the mangrove and then slowly returns to being water based as other lifeforms consume the milk. Given the common proximity of mangroves to the sea, it is possible that it might stay as a milky river until it flows out into the sea but that would be down to the author.
How human-like can insects get before you call what they do "artificial" rather than "natural"?
I envision a city-sized insect colony. The individual insects are very dumb and tiny, just like on Earth, but the colony as a whole is much larger and smarter than any insect colony on Earth.
Just like some Earth insects farm fungi, these alien insects farm mammals and milk them regularly. Somehow they worked around the relative size issue. A river of milk flows from the structure where they milk the mammals, as a simple means of transporting it within the colony.