It's worth noting that milk is, in itself, over 80% water.I'm going to guestimate 85% as an average of the various types of milk. Assuming that this milk-forming operation isn't recursive, and only affects free-flowing water or liquids containing water, we can instead rewrite the question as:
What would happen if 15% of the world's water turned into biological solids?
By biological solids I mostly mean fats, though the milk from various species contain varying amounts of other materials, and we have to account for the sugar and cocoa solids to make the milk chocolate. Needless to say, this would be a bad thing.
For starters: You've just reduced the amount of water available to evaporate into the atmosphere by 15%. The volume of water in the oceans and rivers makes the volume of water in the atmosphere seem like a drop in the bucket (pun intended), so humidity worldwide drops like a stone, causing climate havoc. Oh, and the phrase 'chocolate rain' becomes literal and thoroughly unpleasant for the first few months until the normal water cycle reasserts itself.
Secondly: You've effectively altered the density of water worldwide, leading to current currents changing in a wildly unpredictable manner. Again: Climate havoc.
Thirdly: Any number of marine species aren't going to be able to cope with the immediate change in salinity, viscosity, composition, oxygen content or acidity of their habitat, and will die, choking to death as they try to breath fat, sugar and cocoa instead of clear water. This is bad, and would turn the seas into a singularly unpleasant chowder.
Fourthly (is that a word?): The coastlines/riverbanks everywhere would become huge banks of fatty, fishy deposits as the water in the chocolate milk evaporated and left the solids behind. This rotting mass of sour solids would not be good for any creatures relying on that particular water source, though creatures near fresh water springs or relying on rainwater would avoid this issue. Evaporated water from the milk still forms fresh rain, and aquifers are often pumped up through layers of porous material that would filter the milk back to water, if they didn't clog up.
Despite all of the above: I think humanity would survive. Coastal countries or cities would have the worst of it, with alpine or icebound nations being able to last best (thanks to the ready supply of frozen, nutritious milk in the glaciers/snowbanks) Edit: After seeing Spacelizard's numbers on the melting point of chocolate milk, I'm going to have to revise this a little. Alpine countries would not last well (Nobody likes chocolate flash floods). Countries in the far, far north or at extreme elevations would probably be OK. Siberia would be a good bet as it's high, dry and cold.
After a few months of weather hell the normal water cycle would start to reassert itself, leading to desertification, and leaving behind roughly 800,000,000 metric tonnes of fat, sugar and cocoa solids (humanity weighs roughly 290,000,000 tonnes). This would provide fresh water for the remaining humans/land species/sea species that can survive milk (Looking at the snails , worms and bacteria mostly), along with a vast nutrient bank for any remaining vegetation.
Scroll forward a few hundred years and a lot of the milk would have been recycled by the biosphere, leading to a very dry but also nutrient rich world. The few areas that aren't desert would be lush oases, where the new mankind would never, ever look at milkshakes the same way.