We received a complaint that an adult employee is dating a minor he met when she was registering to obtain medical care - who do I file this with for further investigation?

Who handles investigations for your employer? Human Resources? The Supervisor? A medical Ethics Board of some kind? There is no one the law dictates you must report it to. It is up to company policy.

I am under the impression that this became a sexual relationship - which obligates me to report it to the authorities -

I have no idea what your relationship is to this couple, but unless you suspect this child is being abused, it really isn't your place to get involved. You don't know it is sexual and depending on the ages of those involved and what the age of consent is in your state, it may not even be illegal.

If this person somehow violated a work related policy involving fraternizing with patients or former patients, report that to the appropriate person at your job. Anything else really isn't your concern. If this is a young child in danger, that is different. If this is a coworker that is dating someone under 18 consentually, I'd stay out of it.

I am under the impression that this became a sexual relationship - which obligates me to report it to the authorities -

That may or may not be the case, but if it is, it is not an employment law question. You might have better luck getting an answer on the family law or the criminal law boards.