I never thought of that one. It's fun to think "we know the whole universe isn't spinning very fast, because our gyros are stable". Feels both obvious and somehow bigger-than-life to me.
We are not kinetically bound to the galactic center, there is no friction causing earth to remain "upright" in respect to the galaxy. Earth is also a freely rotating inertial body and, even though wobbly, it is itself a gyroscope.
The next level of stabilization would probably be gravitational waves.