The answer to this has much, much more to do with the U.S. Army than it does with your specific residency program or the ACGME.
You are under orders, or you will be soon, to report to your duty station by a certain date. Don't violate these. Doing so can have significant legal (read: UCMJ) consequences.
Assuming that you are still not well in a few weeks to months, the simplest way forward is to report to your residency program and then seek out the medical attention you need. This is simplest because you will be active duty at that point, meaning you will have ready access to treatment that is in the 'system'. The Army isn't going to let you delay anything unless everything is documented, so getting it into your medical record is an important step. You should have a decent amount of time to do this, as military residency orientations tend to be relatively long. If you are deemed to be unfit for duty, then you can delay start of residency. The good news for you is that off-cycle residents are very common in military GME.
By the way, this assumes that this is a temporary condition from which you can completely recover. If this event has resulted in more permanent changes in your health, then - over the long haul - that alters the calculus completely, and you may not want to wait until on active duty to report these changes.
If you feel like you cannot report as scheduled, then you need to have your orders amended. Assuming that you are HPSP, this means contacting Falls Church and finding out what you need to provide to justify such an amendment. This process has a lot more moving parts and will, accordingly, be much more complicated.
Irrespective of which route you take, make sure you 1) keep your program director informed and 2) keep copies of your civilian medical record. Regardless of how this plays out, you're going to want this event to make it into your military medical record, and the most reliable way to make that happen is to have the records in your possession. That's good advice for all important military documents, by that way. That is, maintain your own repository of critical documents, because the Army will absolutely lose things and you will be SOL unless you can provide the relevant forms.
