

Being high doesn't put you in orbit, it takes velocity relative to Earth to do that.

(Rockets going into orbit aren't just climbing, they're also building ground-speed. If there were an imaginary tower from ground to low-earth-orbit height, climbing it wouldn't put you in orbit - if you let go of something at the top of that tower, it would just fall.)

Unless the tower reaches to geosynchronous orbit (~36,000 km altitude). Then you've got a space elevator.

You do get some relative velocity due to the earth's rotation, it's just not enough to put you in orbit until you reach geosynchronous altitude (from your imaginary tower, not necessarily the imaginary sphere of water).

It's still not enough. At the equator, to do one revolution per day, you're doing ~1000mph. At 36,000km, because the circumference of the orbit is much longer than the circumference at ground level, to do one revolution per day, you need to be doing ~6900mph. So even if you get to geosynch, you still need to pick up ~5900mph sideways in linear velocity. As well as the energy it takes to get up there.
