Is all vision based on movement? This is something I recall from a class on human perception: vision is based on "movement" or change in what hits receptors, so if your eyes were perfectly still you couldn't see, but always-present minor head movements are enough to allow you to see even when objects and you are "still", or at least what is perceived as still.

So is this true? I kept google searching but couldn't find a formal source for this, but I'm not sure if I'm simply missing the term to search for. I also can't find my textbook which, I assume, covered this.
No.

Certainly, in case of artificial vision we know that useful information can be derived from stagnant stimulus on the receptors. Where upon the spatial density of the receptor elements is uniform across the sensor.

Saccadic eye movements make up for the fact that the receptor density especially for color vision falls off sharply as the angle of view increases. So we can view color and sharp detail in a very small field of view. But, intuitively, within this cone developed minds do perceive detail and color since they have special neural connections that can fire if details such as edges of certain angles and thicknesses fall upon them. Also, intuitively, outside of the field of view receptors should be able to perceive details low in spatial frequency.

A quick google on 'vision without saccadic eye movements' revealed This paper which mentions a subject without the ability to move her eyes, who could read with amazing ease. However, she used her head movements to make up for it. One way to know a real answer to your literal unqualified question would be to do an experiment with such a subject and see what she can perceive without head movements. 

One other way to see it is mentioned on chap11, page4 from the book referred in the first link in my post:


  Zihl, et al. (1983), reported a patient who had a stroke that damaged a particular region homologous to area MT (MT is in the monkey brain). This patient could no longer see motion although her vision was otherwise normal. This is known as motion agnosia.


Certainly we are not born with developed visual neural pathways and how changing stimuli and feedback help us develop them with time is being studied actively such as in Project Prakash