First off, the ACA requires insurers to compensate health care providers for "emergency services" at the in-network rates even if they're out-of-network. "emergency" is defined using the "prudent layperson" standard; if you were an average intelligent individual with no specialized medical training, who upon seeing your injuries thought that you would die or be severely disabled without the care, it's an emergency.
If that is the case, the insurance company will pay the in-network percentage of the bills, after deductible. Usually, they are only required to pay the in-network percentage of the amount they typically negotiate with in-network providers, which can be a significant discount. Because the provider's out-of-network, they have made no agreement to be satisfied with that payment, and can balance-bill you for more than the matching coinsurance you'd otherwise need to pay to an in-network provider.
However, it's doubtful that "emergency care" would fly given the information you provided. A broken finger is not a threat to life, or even permanent disability. Even if the finger was sideways, as long as the bone wasn't showing through the skin, anyone with a general first-aid class under their belt (which is not "specialized medical knowledge") would be able to set and splint it well enough to get you to a visit with your PCP. You could argue for checking out the head wound, but I doubt that will fly either if you never lost consciousness or became incoherent.
In that case, insurers will state, in their summary of benefits, the percentage of costs that will be paid to out-of-network providers, if any. They will also send you an explanation of benefits, detailing what was charged, what the insurer paid, and what's left over. First off, the provider will charge you full price; because they're not in-network, they've made no agreements with the insurer to reduce costs for insured patients. Second, you will likely have to meet your deductible before the insurance company pays anything. I've seen and had employer health plans that required deductibles of up to $5,000 before coinsurance kicked in. I've also seen plans that required higher deductibles when out-of-network care was involved.
So, depending on your deductible, how much you used your health plan prior to that event, what the hospital billed, and what your insurance paid, you could owe, as littleadv said, between a few hundred and several thousand dollars. Ouch, my friend. Let this be a lesson; if you weren't hurt on the job (for which workman's comp typically applies no matter where you go) and it's not so bad that you're forced to take the first available option and can't wait for an in-network provider like your PCP or at least a CareNow, then stay in-network even if it means gutting it up a little more in the interim. Even an in-network hospital stay can be expensive; I went in once for a broken collarbone, and had to pay the ambulance service (city-operated), the hospital copay, a separate copay to the doctor who saw me, and because I didn't meet the deductible, two X-rays to the radiology office which was co-located but not affiliated with the hospital, all separately.
One last thing. Some sports leagues require all participants to purchase supplemental insurance. I had to do this for hockey starting the season after I broke my collarbone. This is typically pretty cheap, like $35 for the whole year, and basically pays out for medical bills above and beyond what's been covered by your primary insurer. You may have this coverage paid for as part of your entry fee into a sanctioned league. Look into it; you could save thousands.