It's in the US, don't worry
Thanks for your help!
In the US, emergency rooms can't refuse you, whether or not you have money, insurance, or even ID.
Take it from me, I was uninsured for fifteen years, and on the few occasions I got really sick, I'd just go to the ER.
You can always just say, "I'm uninsured, I don't have any money with me, I don't have ID with me today, my name is _______ (fake name) and my address is _______ (fake address). Just bill me."
Or, alternately, give your real name and address, and they will bill you. And then you can either pay it, or not. If you don't pay, they can't refuse you service the next time.
Anyone who doesn't believe this is free to go try it.
An uninsured minor, however, would probably have Medicaid if her parents couldn't afford to insure her. Also, each state has a special low-cost or sliding-scale option for insuring the children of the working-class poor, who make a little too much to qualify for Medicaid.
It's really not realistic that a minor in her father's care would simply have no insurance, unless he lost his job and health insurance benefits the month before and hasn't had time to apply for Medicaid or CHIP yet.
As far as the knife wound; I don't see why it would render her unconscious, unless she passed out from emotional shock or something.
It's really not that easy to make the average person pass out (and stay out long enough to be transported to ER).
A superficial knife wouldn't be painful enough to do it (trust me; I've endured natural childbirth without any painkillers, not to mention stitches in my chin, a broken arm, and two kidney infections, and not come close to passing out, although I might've wished I could).
I don't think she'd lose enough blood to pass out, either.
It really doesn't ring true, given the injury you describe, that she'd pass out at all. People endure terrible trauma every day. Car accidents, etc. Unless it's head trauma, they do not typically lose consciousness unless they are actually near death (from blood loss, etc). This knife wound doesn't sound anywhere near serious enough to make a person lose consciousness.
She'd probably get a tetanus shot and some stitches (they might use tape or glue instead, depending upon the depth of the cut).
Child Protective Services would probably get involved.
A social worker would almost certainly speak to her alone, out of her father's presence.
Nothing her father had to say would make any difference; only
she would be able to convince them that abuse/assault had not occurred. If she denied it and stuck to the "dropped a kitchen knife" story, they'd probably be forced to accept it even if they suspected it wasn't true, because of her age. If she was seven or eight, they might remove her from her father's custody regardless of what she said, but at sixteen they probably wouldn't, if he and she both denied any problem and if their stories were consistent.