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Oxford Learner's dictionary says, in its definition of on:
5 immediately after something
On arriving home I discovered they had gone.
There was a letter waiting for him on his return.
But I think it's more like 'when' than 'after', isn't it?
What does, for example, "On returning from holidays" mean? After arriving home? Can't this also mean "On the way home" and even "When about to get going home"?
On [some event] is frequently used to add an element of drama to a sentence, to imply that one event happened after another with no time for preparation or thought. As Catija's comment says, "on arriving home" can mean the same as "upon arriving home", which is to say, "at the moment of arrival". Example:
On seeing the roach she grabbed the can of bug spray.
As P.E. Dant's comment points out, it can also be an imprecise statement that means soon after some event, where the exact amount of time is not important:
On their return they found waiting for them a mysterious package addressed simply, 'To the residents of this house'.
I've found an example.
"On opening the box, be prepared for a slight surprise, as the kit has been (...)."
"(up)on handling" or "(up)on delivery" mean "when handling" and "when delivering," and therefore they are ambiguous, needing context to be clear.
"On arriving home, I noticed a new store had been opened nearby" means just before you arrived home. You can say this before actually enter your house. This is not because 'home' means hometown, but because 'on' doesn't have any sense of 'after.'
So, this statement
"On returning from Rome, he wrote to the ~ (This indicates that he wrote whilst he was in Rome)"
is indeed correct. It can mean he wrote when he was about to leave Rome. I do trust this writer's sense of language as much above average.
on the occasion of
(formal) at the time of an important event
on the occasion of his second wedding
from LongmanSynonyms: at the time when, (...)
from WordReference English Thesaurus
More: on
'On' has the sense of 'upon,' 'toward,' 'forward,' 'near,' but not 'after.'
