What grammar rule makes this sentence incorrect:  "Not until did I go
home, I realized that I had left my keys at the office."?  -namely,
the 'did I go, part -mustn't it read: Not until I went home, did I
realize that I had left my keys at the office"?

Isn't the past term of the verb 'to go', 'went'?
I would write 'It was not until I went home'

Alan,
I think you are absolutely correct with your second version, although
I cannot name a grammatical rule.
I could imagine a situation in oral communication  - a context -  in
which one might want to use "did" for emphasisze in both phrases: 
"Not until I did go home, did I realize that I had left my keys in the
office."  But that would be special case, after having mentioned that
you had left the office to go somewhere, and then later, on the way
home, remembered the keys.

Not much help, I guess.

Curious
"Not until did I go home"      - is rather Germanic

Not until I got home           - more British

It wasn't until I got home     - more likely British

It wasn't until I had got home - more vernacular

Not until I had got home       - possible

'got home' is not a verb - it is a 'state' more of an adverb
'at home'  is similar - but that implies inside (keys Ok - but wallet missing)

I think the problem is mixing two verbs 'did' Imperfect for 'do' and 'go home'
- 'go home' is distinct from being 'at home'
- to 'go home' is an action, but ones physical location is indeterminate

My grammar is more received than formal, I only really learnt it
studying French and German.
However the 'received' patterns are laid down pretty well.

Digressing, I once tried to explain the 'furtive' case to a translator
level German.  She found it hard to believe that 'they' could be
singular.

Your sentence is a run-on sentence
substitution may be....


I didn't realize I left my keys at the office until I got home.



I'm not a college grad, but it makes sense to me....lol

Usually I need my keys to start my car. :P

Well, you can say this sentence more efficiently by simply saying, "I
realized that I had left my keys at the office when I went home." 
But, for emphasis, you can start a sentence with a negative word
(never, little, Not Until, Rarely) and then you must invert the verb,
as if it were a question, even though it's not a question.  Usually
you invert the first verb.  It is clear to me that "Not until did I
go..." is incorrect.  I just don't know why, grammatically.

examples of this inversion:
Little do I know about China.
Rarely do I travel outside of America.
Rarely have I traveled outsied of America.
Never have I been to Chicago.
Not until I saw (not inverted) the movie, did I know (inverted) the ending.

Why not "Not until did I see..." ?  My guess is it's because "not
until I saw" is a dependent clause, so you must invert the first
independent clause, which is "I knew the ending" --> "did I know the
ending."

still I'm not sure...thanks for your help!