He is not moving apartments because he hasn't got a truck that is big enough to move them....
how would you say "he is not moving out of an apartment"?
I think this sentence was about moving INTO an apartment.
Of course, if such a distinction is never made in Hebrew, then I understand.
This sentence is about moving between apartments, from one and into the other. The concept is quite similar to הוא מחליף חולצה, "he is changing his shirt". If you want to say he is moving out, you can say הוא עוזב את הדירה or maybe הוא עובר מהדירה, but that is less frequent. But you can say הוא עובר לדירה, he is moving in.
Can't you also say, "he does not pass an apartment?" The sentence "he does not move an apartment" makes no sense; it's moving into or out of an apartment.
The normal way to express this in English is "he is not moving", and context tells you that this is about going from living in one place to living in another, rather than about standing still like a statue.
If you said your sentence to an English speaker, they would have no idea what you're saying.