I'm just trying to figure out if "neither I" is grammatically correct as a standalone statment (in spoken English).
No, it is grammatically unacceptable.
Correct constructions using neither include:
Neither a nor b: This is the most common usage.
Neither Alaska nor Florida charges state income tax.
Neither noun: You can use this variant when it's clear what you are referring to.
I moved from Alaska to Florida this year. Fortunately, neither state charges income tax.
Me neither: This may be close to what you had in mind.
"I don't like Brussels sprouts."
Neither: This could be a one-word sentence.
"Do you prefer carrots or broccoli?"
As I said, I already heard "Neither I". If it would be grammaticaly incorrect indeed, this is the contraction for "Neither (do) I" where the verb is implicit like in : "he's faster than I" for than I am...
We also have :"Neither I nor Bruce could handle it" where a third part would also say : "Neither I" for "Nor I" or "me neither" or "me either I could not" BUT NOT "me either I could" to mean "could not" because with the negative "either" can't match here...
But we've worst like : He DO or "my baby just CARE for me" in the song...