If I take the longest hose you’ve ever seen, put one end in the Atlantic and the other end outside the atmosphere, then use a pump long enough to start sucking water into space, would the pressure differential keep the water flowing when the pump is shut off, thus emptying Earth’s oceans?
Nope. Because the pressure differential is exactly counterbalanced by the gravitational differential.
I should add that the above is also the reason the atmosphere doesn’t flow away into space despite the pressure differential between ground level and the surrounding vacuum of space.
That’s what I suspected. So you would pretty much have to put the other end of the hose on a heavier planet than Earth to empty the oceans.
crosses out another item on the list
Your water column will separate once it gets 32 feet above sea level.
To pump up to where the atmosphere ends, you need to pressurize the water at the bottom of the pipe.
Not necessarily - apparently, an unbroken column of pure water has tensile strength and can be pulled higher than that - although probably still nowhere near the top of the atmosphere.
Seawater isn’t anywhere near pure.
I’m puzzled by this.
The height at which you’re “outside the atmosphere” can be a subject of dispute, but at the modest height of 100,000 ft (which puts you above 99% of the atmosphere) the ambient pressure would be around 14.6 psi lower than at sea level. By contrast, gravity operating on a column of water 100,000 ft high would produce a pressure at its bottom well over 40,000 psi.
Which is why you can’t have an unsupported column of water 100,000 feet high. I’m not sure where your confusion lies, frankly. Can you expand a little?
A vacuum will only suck water 33 feet against gravity. You can’t suck harder than a vacuum. A siphon will not pull water over 35 foot hill just as it won’t pull it higher straight up than about 33 feet.
With enough pressure, which is not limited as vacuum is, you can push water much higher. That takes power.
Can’t siphon it off the world because you have to go over 33 feet on the earth first and ::: ‘everyone’ ::: a vacuum will only pull to about 33 feet.
YMMV
The OP’s question was: would the pressure differential keep the water flowing when the pump is shut off?
Your answer was: Nope. Because the pressure differential is exactly counterbalanced by the gravitational differential.
Whereas it seemed to me that if a 100,000 ft column of water were somehow created, gravitational force on the water would not exactly counterbalance the pressure difference - it would massively exceed it.
Get creative. Now if you put the pipe end at the bottom of one of these massive whirlpools can they shoot the water into space? Whirlpools are siphons too.
Well, the water column would settle down into an equilibrium state where the gravitational force and the suction would exactly counterbalance. That would involve the water column collapsing down to about 33’ high, as stated.
The point is that a vacuum doesn’t suck - ambient pressure blows. All the vacuum does is remove the opposite-direction blow that is the default condition. But this still only gives you about 14 - 15psi of blow - not enough to push water very high. You cannot improve on that no matter how good you make your vacuum; all you can do is use some kind of pressure other than atmospheric (which is why, to extract water from a deep well, you need a pump that pushes from below, not one that pulls from above; at most you can only pull it for the last ~30 feet (less in most practical applications) ).
Get a big sheet of Cavorite, though, and you’re good to go. Apart from a few other practical considerations which I here leave as an exercise for the advanced student. 
Then how do you account for the success of siphon aqueducts? The siphon on the Los Angeles Aqueduct at Jawbone Canyon draws water out of an 850 foot drop:
More info on Jawbone Canyon siphon, with pictures:
This was once the largest siphon in the world, though other aqueducts may have surpassed it since.
Though not stated by Q.E.D., hence the possibility of confusion.
Saying something along the lines of “the pressure differential would wind up exactly counterbalanced by the gravitational differential” would have helped. Saying “is exactly counterbalanced” seems to imply that the pressure in a 20-mile high column of water could be balanced by 14.6 psi.
Ah, sorry. I see now where the confusion lies, and the error was mine in not being clearer.
Spoken like someone who’s never seen Die Hard 2.
Bawahahahha How very true… 
You will note that no where along it’s length does it get to a point 33 higher than the starting elevation. It can drop 800 feet and go back up 820 feet and then drop 870 feet and still work. BUT… if it tries to go 33 feet above it’s starting elevation, the siphon will break and not flow…
YMMV