Let’s say, somehow, you were able to cleave the planet into two halves, would the oceans fall toward the core? Would the water float or get flung off into space? I imagine if you had the energy to break the planet into huge chunks, the water would just go everywhere, but I was wondering what would happen if you could suddenly force two halves to separate.
The whole thing is liquid enough to slosh into a glowing mess and evaporate all the water into smoky clouds. I doubt there would be any sizeable solid bits on the surface.
Some people will do anything for Beach-Front real-estate.
The oceans would fall toward the core… but so would everything else. A magically cleaved Earth would be incredibly unstable - the core is liquid and the outer mantle is at least a sort of plastic sludgy consistency. Each half would collapse into a spheroid (assuming that your magical cleaving method also counteracts gravity and prevents the two halves from recombining).
The water would certainly fall down as the new spheroid forms, but the rest of the Earth, including the crust, would be flowing/falling at just about the same rate as the water.
Since the inside of the Earth is so hot, the water would be converted to steam. This, combined with massive amounts of gasses dissolved in magma, would form a new atmosphere that would be pretty nasty from a human perspective. Eventually the water would rain out with the sulfur compounds as acid rain and then start to form new oceans.
For one thing, don’t overestimate the amount of water in the oceans (plus lakes and rivers), relative to the total volume of the planet. In the OP’s scenario, everything in each half of the planet would collapse toward the center, and would eventually reform into a very different planet (or two), probably with no liquid water at all. The biggest variable would be how the two halves were separated, and how fast and how far.
Isaac Asimov said to visualize it this way: if the earth were a basketball-sized sphere of room temperature steel – and you breathed on it, once – “Hahh” – like polishing your eyeglasses – the sphere now has as much water on it, proportionally, as the earth has oceans.
This has already happened, in the Earth’s past, after a fashion; according to the Big Splash theory, a Mars-sized object hit the primeval Earth, which was itself somewhat larger than it is today. The impact was messy, and very, very violent indeed, but the eventual outcome was that the material of these two objects was mixed up and redistributed somewhat, so that the (now-slightly-larger) Earth was orbited by a mishmash of very hot objects that eventually coalesced into the Moon.
How does this resemble the situation in the OP, one might ask? Well, consider the pre-impact Earth, which probably had at least some water content. When the impact occured, almost all the things described by dracoi actually happened to the water on the Earth; the water was flung into space by the impact together with red hot magma and magmatic gases, and only a (rather uncertain) fraction of that liquid fell back to Earth to form our oceans.
Some of the rest was incorporated into the red-hot Moon swarm, and some was simply blown away by the solar wind and light pressure. At the end of this process the Earth/ Moon system was probably quite a bit drier than it was before the impact.
I have always liked this picture to help visualize our planet’s ‘water -v- solids’ proportion. Might help a bit 
Here’s a neat picture along those lines, from the United States Geological Survey. There’s three spheres representing total saltwater, total freshwater and water from lakes and rivers.
ETA: Ninja’ed by Ionizer!
You mean separate and then keep them apart? Because if you’re not using your magical powers to keep them apart, they’d smash right back together because of gravity.
And, as mentioned above, if you’re holding them apart magically, are you shoring up the brittle crust and molten upper layers to keep them from falling into the gravity center before the water cascades in?
The same thing that would happen to any other matter in the planet’s bulk that is fluid, such as lava and molten metals… It will flow according to the new gravitational forces that are exhibited by the new bodies that are no longer unified into a single larger planet.
Presumably, though, great heat will be released by the exposure of lava and other hot metallic liquids which will quickly evaporate most of the water, and it will then behave instead like the other gases associated with the planetary fragments.
All the water would run out of the ocean basins and out the bottom of the galaxy.
I have too much respect for Dr Asimov to suggest that he’s wrong, so instead I’ll ask where my mistake is:
Here’s my calculation: According to Wiki, the average depth of the world’s oceans is 3,682 meters, and covers 72% of the world’s surface, so let’s take the averaged-out average to be 2.65 kilometers. Compared to the earth’s mean radius of 6371 km, that’s about 0.0416%.
I was surprised to find that the Wiki article on basketballs describes their size by circumference rather than radius or diameter. Okay, whatever. The circumference given there is about 749 mm, so let’s use a radius of 119 mm.
A layer of 0.0416% on this basketball would be 0.0496 mm thick. Let’s compare that to the thickness of a plastic bag. From http://ziplock-bags.com/ziplock_bags_faq.html
Thus, if the earth were basketball-sized, then the oceans would be about 0.0496 mm deep, which is noticeably thicker than a standard household freezer bag, and MUCH thicker than a sandwich bag (though slightly thinner than an industrial bag). And compared to the moisture of breathing on such a ball, I gotta wonder where Dr Asimov got that comparison.
Keeve, looking at the picture in Pine Fresh Scent’s link, there’s clearly more water than a single breath would hold, so I’d guess you don’t have a mistake. Perhaps Asimov’s statement originally said fresh water (the medium sized water sphere). Even that seems like too much to me, though. Maybe a billiard ball-sized sphere?
The question is though, how much moisture in in a breath? This page says up to 500 milliliters a day, which is over an average of about 15 breaths per minute or 21,600 per day, for a water loss of 0.023 ml (cubic centimeters) per breath. According to Keeve’s post, a basketball has a radius of 11.9 cm for a volume of 7,059 cm^3. The ratio is thus 7,059:0.023 or 306,913:1; for the water on Earth, the volume of Earth is 1,083,206,916,846 km^3, which is a ratio of 781:1 (using the volume of water given by the USGS in Pine Fresh Scent’s link).
Thus, Asimov is indeed wrong - by a factor of 393 too low. Also, per ZenBeam’s question about the sphere being the size of a billiard ball instead, a billiard ball has a volume of 97.7 cm^3 (American size, 57.15 mm diameter), for a ratio of 4,249:1, so that is also still too low (note that both ratios for exhaled breath assume 100% condensation, even an ice-cold sphere won’t obtain this).
Or, as **ZenBeam **suggested, he meant fresh water only (about 2.5% of the total on Earth, or a ratio of about 31,240:1, which looks plausible for a billiard ball, including a realistic amount of condensation, but still an order of magnitude too low for a basketball even with 100% condensation).