Why can’t my eyes gaze around at a constant rate?

Short answer
Smooth-pursuit eye movements require something to pursuit. Intentionally attempting to make such an eye movement in the absence of an appropriate moving visual target will result in a saccade instead.

Background
Your question concerns two types of basic eye movements, namely saccades and smooth-pursuit eye movements.

Saccades are rapid movements of the eyes to change the point of fixation. They occur for instance during reading and while gazing across a room. Saccades can be made voluntarily or involuntary (Purves, 2001).

Smooth pursuit movements are slower eye movements that keep a moving stimulus focused on the retina. One can voluntarily decide whether to follow a moving stimulus or not. However, making a smooth-pursuit movement in the absence of a moving stimulus is next to impossible for most people. Instead, when attempting to do this, a saccade is made instead. Performing a smooth-pursuit movement in the absence of a moving target seems to be possible, but requires a lot of training (Purves, 2001).

<sub>Reference
- Purves et al. (eds). Neuroscience, 2^nd ed. Sunderland (MA): Sinauer Associates (2001). Types of Eye Movements and Their Functions</sub>