Can a rehab facility discharge you early for refusing hospitalization?

"... she didn't feel it necessary [to go to hospital] just because she was nauseous."

Mm. Most people wouldn't go to hospital "just" because they felt nauseous after eating a meal, perhaps a meal that disagreed with them in some way.

But your mother was in rehab for alcohol abuse. Nausea in a person who's being treated for alcohol abuse could be an extremely important clinical sign - there's no "just" about it - and the rehab unit would have been negligent had they not insisted on its investigation.

So, they've not so much booted her out, as refused to take any further responsibility for a patient who will not allow them to act responsibly.

Yes and it could well pose issues later on.
If her medical chart shows that she left the facility AMA - against medical advice- the next time she tries to use her insurance & if it’s on the same IDC-10 codes as the AMA notation in her chart, insurance may not pay the claim.

I would say Yes as if the client doesn’t feel she needs treatment. They can’t and won’t keep her against her will and won’t/can’t continue to bill insurance if the patient is not showing documented progress that she is getting better. 

Probably yes. A rehab facility operates under a State license and has certain medical protocols which must be followed.

Vomiting could be symptomatic of an upset stomach or something very serious, in the context of detoxing from alcohol. The facility has every right to insist that your mom follow their rules and protocols.