If see is used in the literal sense of perceiving light via the retina, then:
Can tends to express the speaker's concern surrounding optical conditions, acuteness of sight of the interlocutor, the small angle subtended by the target in the field of vision, and so on and so on.
Also, "I can't see a thing" means the speaker is (temporarily) blind.
Do simply asked whether the interlocutor has discerned the object.
"I don't see a thing" expresses that the speaker considers their eyesight to be in order, but there are other circumstances that making visual perception difficult (and which the speaker wishes to be taken care of!).
There, can you see it?
-No, what is it? Where?
The smallish yellow blob, near the crest of the hill.
It just started moving to the right.
Do you see it now?
-Yes, I think I do.