honestly, as much as they would like a patient to stay in their care for further monitoring, they can't. if the patient insists they're okay and once they signed the DAMA or discharge against medical advice then they can go.
I think it depends on the country, then. Where I live, if injuries like these are noticed, the patient undergoes a psychiatric evaluation. If it turns out that they actually intend to end their life (in this case, he was hit by a car, so the question is whether Mokyeon will tell them that, or lie about how he got injured for Nanjo's sake. Since he walked in front of a car while drunk → he could have done it intentionally, especially considering the injuries on his arms), then here we have the option of emergency psychiatric treatment, even without the patient’s consent, provided that the conditions set out by law are met. That said, obviously, I don’t necessarily expect a fictional story to be absolutely accurate in every respect.
So... no one at the hospital noticed the relatively fresh wounds on his arm that suggest he hurt himself and might be suicidal, so even if they run some tests, they still shouldn't discharge him just like that?