Well, you CAN do it, but it's not the best way to do those types of measurements on land. The advantage of a sextant on a boat is that you can account for the rocking of the boat by rhythmically rocking your body back and forth. If you take a sextant, add a bubble level, attach it to a tripod, and remove the mirrors used to line up with the horizon, what you have is a transit or theodolite.
That said, if you have a sextant, you can still use it on land. The best way to do that is by using an artificial horizon, which is simply a pan of water. You line up the reflection of the Sun in the water with the reflection of the Sun through the sextant mirrors. You have to divide the resulting angle by 2, and most professional so-called "sextants" actually go well past 90 degrees to make this type of measurement possible.
If you try to use a level attached to your sextant, you have to calibrate it to make sure it really lines up with the 0 point. With the pan of water, the water's surface levels itself. And if you try this method, you'll find it's actually easier than taking a measurement using the horizon, and you might wonder why not use this same method on a boat. But the rocking of the boat will put ripples in the water, making it tough to get a good reading.
An aircraft sextant, actually does work the way you're thinking. In an aircraft, you usually don't have access to a flat horizon, but still have to deal with the motion of the aircraft, so a bubble sextant actually makes sense in that case. I'm not sure if this method is more or less accurate than a regular sextant, but I can tell you the aircraft sextants are significantly more expensive, and likely require more care and calibration, which is probably why they're not generally used when a horizon line is available.
There's also a device called an Abney Level, which is mostly just a protractor, a short tube, a level and a mirror. And works pretty much like a bubble sextant would, but without the mirrors, so it's a much cheaper device. Calibration is still an issue though, and measuring large angles is difficult as you're looking at the reflection of the level at a steeper and steeper angle.
But, on land, a theodolite or transit are the preferred, and more accurate tools. They are less frequently used though, because once one person has done the measurement, it's not going to change very quickly, and you can just use their measurement.