All that is legally required is stabilization. Which can be very squishy to define. So it’s entirely likely that if this woman was in the USA, with no health insurance and no money, her injured hand would have been stabilized in some sort of brace, and she would have been sent on her way, with instructions to “go to the ER” if it seemed like it was infected.
The impetus for the EMTALA law, which was and is greatly opposed by conservatives, was hospitals turfing women in labor out of the ER if they couldn’t pay, so babies were being born on the sidewalks or parking lots outside.