Yes, "on/upon closer examination" is fairly common. It means you probably looked better at something, or that the appearances may have been deceiving -- until you took a closer look.
"closing" doesn't fit here. You don't "close" the examination, you just examine something more thoroughly.
Thanks for strengthening my 'new awakening'. What still lingers in my mind is whether the expression "upon closing the examination" does not make sense at all.
Here, 'upon' means immediately after, and 'examination' is the same as investigation.
Then, I believe that 'upon closing the examination' is almost the same as 'immediately after they completed the investigation', which I think would make perfect sense.
Well, that's not what the expression means. Again, "close" means that you came closer (nearer), not that something has been closed.
Also, you don't open or close an examination, at least not in the sense of the "upon closer examination". Eg. the bank note looked normal, but upon closer examination it appeared to be a counterfeit. You can't talk of an investigation here; the whole thing took perhaps one minute.