All objects moving in a circle will be subjected to a force pointing to the center of the circle. This force is centripetal force!
Because all points on the same radius are subjected to the same force pointing to the center of the circle, their forces are equivalent to a state of force balance. The balance of the whole gyro is guaranteed.
Newton's first law also holds in this case. Gyroscope keeps rotating balance, and as a whole, it is forbidden to move. This is the so-called system quiescence. This state is relatively forbidden, and you can't simply see that a certain point on the gyro is not static!