Check with the billing/coding department of the hospital as well, they might have been the ones who fucked up.

They will invariably say that it's impossible for them to have made a mistake. It would take them 5 minutes to look it up, but the people in billing departments would rather spend 40 minutes on the phone telling you it's not possible than the 5 minutes it would take them to discover that it's exactly what happened.

Request that the billing department print a copy and send it to you for records. This is the only way I've ever seen people actually get them to check. You'll almost certainly need the information in order to successfully appeal with UHC even if the hospital did everything right.