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I am looking for a word or phrase that portrays the feeling of watching an intimate (but not sexual) moment unfold.

Some context;
A is keeping a watchful eye on E because he is worried about her safety. In the process he realized that he was watching a really personal moment that maybe wasn’t right of him to witness

I know that being a voyeur is generally connected to the sexual gratification from watching - but A is particularly uncomfortable after.

“She (E) closed her eyes, reaching out her arms as if in an embrace and I (A) got the sudden sense of ______”

I think 'intrusion' would work for your sentence:

“Sarah closed her eyes, reaching out her arms as if in an embrace and I got the sudden sense of intrusion”.

The most obvious word in this particular situation would be

Embarrassment - A feeling of self-consciousness, shame, or awkwardness.

In American English, we might call such a person a creeper rather than a voyeur when the primary motivation is hovering nearby or surreptitiously spying private although not necessarily sexual moments.

Oxford Dictionaries:

3 North American, informal A person whose behaviour towards or interest in someone is regarded as unwelcome and socially inappropriate (typically used of a man)

‘I could dance there without a creeper walking up and accosting me’

That would make your sentence

“She closed her eyes, reaching out her arms as if in an embrace, and I suddenly felt like a creeper.”

Your ask for a word or phrase, so the related creepy works as an adjective here or creepiness as a noun.

“She closed her eyes, reaching out her arms as if in an embrace and I felt a sudden sense of creepiness.”

Invasion

I got a sudden sense of Invasion

From https://en.oxforddictionaries.com/definition/invasion

1.2 An unwelcome intrusion into another's domain.

‘random drug testing of employees is an unwarranted invasion of privacy’

There's nothing sexual implied, but the breech of privacy is clear.

Depending on the scenario it could also be:

I got a sudden thrill of Invasion which would be more stalker-ish.

I got a sudden sense of shame

Oxford Dictionaries:

1. a painful feeling of humiliation or distress caused by the consciousness of wrong or foolish behavior.

Seeing that intimate moment made you think it was wrong or foolish of you to be watching her like that. You felt ashamed.