On a globe the size of the earth - some 24,000 miles in circumference - the curvature of water that finds its own level is not measurable by any means we have today, except at distances. Why not? Because the measurement would have to be in angstroms, the size of the water molecules themselves. We have no way to reduce heat in water sufficiently, while at the same time keeping the water liquid, to reduce the motion of the molecules so that we can measure the curvature. Study microcalorimetric functions. However. We know that water curves in a container with adhesion and cohesion. So, we know that water can curve, and not find its own level.
If there is no other container, gravity, cohesion or adhesion, or a combination of the three, will contain the water.
Water does indeed take the shape of an exterior surface. Consider how many people have been wiping their dishes dry after washing them, for thousands of years. Also, dunk a basketball or tennis ball or golf ball in water. Even when the ball is only in the water for a few seconds, the water sticks to the surface of the ball when you pull it out of the water. In other words, it is wet. Turning the ball 360 degrees in a 24-hour period doesn't have anything to do with how dry or wet the ball remains; so, "spinning ball effect" doesn't apply one way or the other.
If you take a partial vacuum system, one which has a gas evenly distributed/dispersed throughout the vacuum, and you place a chunk of solid material within the system, there will be more gas that gathers at the surface of the chunk than there is in any other place in the vacuum system. This means that there is NOT a sudden cut-off of gas. What it means is that there is a gradual lessening of gas as you move away from the chunk.
Globe earth is like this. Ignore the effect of the solar winds for a moment. You need oxygen tanks while climbing at the summit of Mount Everest, because the air is so thin up there that there isn't enough to breathe. The higher you go, the less the air. There is not sudden cut-off. But gravity definitely causes a gradual reduction of air until you reach a point that is near a pure vacuum. Note that in near Earth orbit around the sun, there is still at least one proton ion for every cubic centimeter of space. There is no pure vacuum in near Earth orbit.
Regarding ships and horizons, lie down on the ground, with the side of your head on the ground, and look out of your eye that is nearest the ground. Do this on a flat surface like a sidewalk, but best a flat, straight road. Note where the horizon is. Next, lie down on a table, in the exact same place, but 2.5 feet above the road. See that the position of the horizon has changed. Objects on the road drop off the horizon at different points. You don't really need a telescope for this, but use one if you want. Ships at sea and horizon measurement is stupid... doesn't prove anything because of too many variables. Road horizon at a couple of inches off the surface of the road is a good indicator. Experiment this way and that, and you will see that there is definitely horizon with objects below or partially below the road horizon.