This is a very interesting question. It's also highly speculative, and vague (what does "a doctor... gives them any medicine necessary to protect them from other diseases that might be lurking in their body" mean? Does it mean they treat the symptoms of AIDS (weight loss, diarrhea, etc?) If it does, then the difference between not treating HIV but treating every complication stretches this almost to the point of meaninglessness. If you are postulating treating all diseases, then I do believe this is speculative at best and unanswerable.
The most likely scenario is that the patient would eventually die (but don't we all?) How he would die is pure speculation.
Unfortunately, while you can place a patient in a sterile environment,
you can't sterilize a patient
it's exceedingly difficult to sterilize food
to prevent all possible hospital acquired bacterial infections, they would need a bubble-boy-like unit (very rare and extremely expensive)
you can't predict if some previously unknown virus dormant in our bodies will be able to express itself. (See @Fomite's comment.)
The fact that you stipulate treatment for everything but the virus that replicates in certain blood cells associated with immunity is almost an aside if you're going to treat all pathogens.
Every bacteria and virus he has in his body will go into that sterile environment with him, so it depends what he has going in. When he no longer has enough immune competency to keep these under control, they will take their toll. If you treat him in a sterile environment and give him world-class treatment of all infections (anti-virals would be needed to treat herpesvirus recurrences, like chicken pox and HSV1, HCV, HepBV, or any other number of viruses one can carry, treat the encephalitis that comes with HSV/etc. He has a small chance of possibly outliving the disease, only to die in the 3-5 days after you turn him out, unless you include bone marrow transplant in the treatment as well.
The progress of HIV to AIDS is variable. Less than 0.5% of patients never develop any sign whatsoever of the infection except for undeniably positive titers. What if your patient is in this small but not-nonexistent subtype? Then there are almost everything in between, with fewer patients on the healthy end than the sick end.
My best guess: your patient will die of some ultimately unconquerable disease you are treating in the bubble, for example, multi-drug resistant C. diff, Kaposi's sarcoma, Mycobacterium avium-intracellulare, Toxoplasmosis (Up to 50% of the world's human population is estimated to carry Toxoplasma) liver failure from rampant Epstein-Barr, or some other opportunistic infection from something he carried in there but is either devastating or drug resistant.
You're asking if the virus will just "die" out, effectively curing the patient. That's kind of like letting the heart and lungs die, but keeping the patient alive on a heart-lung bypass machine. That's not a cure, and upon turning it off, the patient will die.
And that's purely a guess, which, on SE, makes it worthless.
Long-term Nonprogressive Disease Among Untreated HIV-Infected Individuals: Clinical Implications of Understanding Immune Control of HIV