Some people would just write a check, but in cases like these it's best to talk to both the provider and to the insurance company to confirm that they're talking to each other; Make sure the provider sent the claim in to the insurance, and make sure that the insurance received it and processed it.
Providers generally have multiple prices - one (much higher) price that they give to people without insurance, and a negotiated, contracted rate that they give to insurance companies. It's possible that the $2200 price they billed you for was the non-insured rate, but that amount was adjusted once your insurance company had finished processing the claim and let the hospital know how much your share would be.
The reason providers charge people without insurance more is because there is no guarantee that they'll ever get payment. They might luck out some percentage of the time (maybe 50%, maybe 80% - I don't know what the real numbers are) and have someone who is willing and able to pay off their full bill, but all too often they just have to write it off as a partial or complete loss.
With insurance, they give a discount largely because they're guaranteed payment for services rendered. If you talk with the billing department of the hospital and tell them you don't have insurance but want to make payments, they'll almost always work with you to lower the price to near insurance levels, just so they can recoup some of their expenses.
You should be getting an EOB (explanation of benefits) from your insurance company that will break down exactly how much the provider billed, what the allowed amount was (the contracted rate for that particular service, procedure, and/or diagnosis), how much the insurance company paid, and how much your share was (deductible and coinsurance).