Looks like Faye lied to her doctors about what's going on, because if she responded truthfully to a basic history regarding alcohol use, she would be getting admitted a psych ward.
I think if getting stuck into a psych ward is one of the usual outcomes when visiting the ER after alcohol abuse, a lot of people would be afraid to seek help.
I would have thought she emptied her entire supply of booze by now.
So, there are two possible scenarios here: either what Marten saw in strip 2881 was vomit made of booze and pills, or it was vomit made of booze and food. Let's take each in turn.
Booze and pills: given history of suicide attempt and mental health issues, it's malpractice not to consult psychiatry in the ED. Given history of suicide attempt and unclear history of intentional or accidental overdose, it's malpractice for the psychiatrist not to recommend a medical hold followed by inpatient psychiatric care. So assuming no malpractice, Marten didn't mention seeing pills (maybe he was too freaked out to give a good history), and Faye minimized what happened sufficiently for the ED doc to make medical decisions without thinking about a pill ingestion. Still bad form not to get a toxicology screen on somebody found down and brought to the ED, though it looks like Marten did it by private car instead of calling EMS, so more understandable. So the medical decision making probably played out more like scenario 2:
Booze, no pills: Faye has been drinking pints of hard liquor daily for what looks like a week, minimum. She drinks more than intended, she drinks when getting up in the morning, she drinks enough to get fired, she drinks enough to get brought to the emergency department. At minimum she meets criteria for a moderate alcohol use disorder, which should make the ED doc suspicious that whatever amount of drinking she endorsed to him or her is less than what she is actually drinking. Alcohol withdrawal can kill you (~5% in-hospital mortality, up to 50% at a year if untreated in some studies), and is absolutely an indication for admission. The only way she gets out of the ED without medical admission for alcohol withdrawal is if she states that she wants to continue drinking (does not appear congruent with her attitude in 2885), at which point she definitely meets criteria for a severe alcohol use disorder.
Marten may not have known about her getting fired, etc., and may not have been able to provide that history. Assuming no malpractice, Faye didn't tell her docs when they asked about it. If she had been admitted to the hospital, the history taken for the admission should include a psychiatric history, which should dig up the history of suicide attempt and big pile of recent psychosocial stressors. Now we're back in "was this a suicide attempt?" territory, and psychiatry should be consulted, though depending on the answers they get from her friends regarding safety/support, she might get linked with outpatient resources. There are some docs I know who might not consult psychiatry if she stated that she was not interested in talking to a psychiatrist, though.
So, if she's getting discharged from the emergency department, she did it through a combination of lying/evading questions that should come up in a basic history. If she got discharged from an inpatient hospital bed (which should be 3-7 days after admission, depending on what happens on the withdrawal front), and there were no pills, I can maybe see a way she gets discharged home instead of to the psych ward.
HOWEVER, I'm pretty sure I see a bunch of pills in the vomit in 2881, so I don't think Faye told them what actually happened, and Marten somehow forgot to mention the pills. If Faye has had been to a psychiatric hospital before (esp. involuntarily, as happens after some suicide attempts), then she may have decided to minimize her history in order to avoid a second stay. This is pretty common behavior.