It might not kill you in five minutes, but if you don't get treatment, it would eventually cost much more.
All you need right now is probably antibiotics and whatever treatment the actual infected area needs. That is something they can do for very little money.
If they turn you away, you say, "If you don't see me now, this is going to get worse, and in a while it will cost a whole lot more for you to fix this when it gets bad enough that, legally, you have to."
Ask for a payment plan, a sliding-scale fee. If you can get a credit card you might be able to put the bill on that--it's not something I'd recommend, but we're talking necessary medical care here. Make it plain that you are trying to negotiate, that you need medical care and you would not be asking if it weren't absolutely necessary.
Ask to talk to a social worker. Hospitals have those. Sometimes, if you can't pay, they can find ways for you to get medical care. Sometimes there are free clinics. If the information they give you doesn't pan out--like they say there's a free clinic but it turns out there's a six month waiting list--you go back to them, explain what happened, explain you need help sooner than that, and ask for alternatives. When they have nowhere to send you, ask them who else might be able to help you. Don't let them go until they've given you some sort of useful information. They don't get to shrug you off; they've got to do their jobs.
Don't give up, and don't get angry. Just be quietly persistent. Sooner or later, they will give in. They won't immediately throw you out because you're not making trouble; you're just constantly *there*. It'll wear them down eventually.