He likes to watch her get hit.
He likes to watch her gets hit.
No matter how much I research, I still can't explain simply why get is correct and gets isn't.
Here's what I think, though. It's a combination of issues:
get hit is having gotten hit in present tense.
It's passive, because we don't specify the agent that is hitting her.
"The subject of a sentence or clause featuring the passive voice denotes the recipient of the action (the patient) rather than the performer (the agent). The passive voice in English is formed periphrastically: the usual form uses the auxiliary verb be (or get) together with the past participle of the main verb.
So the get here is definitely acting passively.
Ignoring tense: get hit ~= be hit ~= being hit
So, the key phrase in question is in passive voice (get is a helper verb), and get hit similarly acts nominalized.
get hit here can be substituted with beatdown:
We're watching (that someone received a) beatdown.
We're watching (whose?) beatdown.
In either case, it's still an argument between her being a direct object or acting possessively, so that's why it can't be she.
And because you can use any of them, her, him, it with get (and never gets in this context,) it just is.