There is no state that requires you to show ID to obtain medical care. To the extent that this is done it is done at the behest of whoever is paying for the care to determine that you are someone who is authorized to benefit from this payment, and not an imposter, or as a matter of policy of the doctor.
(An exception to this general rule applies when one wants to pick up a prescription for a controlled substance or a Sudafed product, where you must indeed show ID to show that your name matches a prescription or to insure that the right name is entered into the Sudafed database.)
It might be more convenient for the doctor in terms of collection of bad debt, insurance policy claims (where an insurance card would normally be required, at least), protection against fraud claims from an insurer, and medical record keeping to have a name, so a doctor might make it a policy to require ID, but it is not required by law (except where a government benefit provider like Medicaid or Medicare is involved and has a regulation requiring it).
For example, in the Las Vegas shooting, where there wasn't time to process paperwork, hospital triage officials simply assigned an alias to every incoming patient and wrote it in marker on their body to keep the medical records straight, and to allow that alias to track medical costs to be billed when the time came to get the proper intake paperwork filled out and the file sent to the accounting department.
There are also other circumstances that do not involve emergency treatment (e.g. STD testing, methadone treatment at free clinics, and clinical trials) where an alias rather than a true name is sometimes used to keep track of patients.