Can player receive Gray Death treatment?

Solution 1:

This is not related to Gray Death. This is for access to use special part of Clinic with MedBots.

See http://deusex.wikia.com/wiki/Free_Clinic for more details (inlcuding 2k price with 40% discount)

Now about (not so relevant) Q in title:

Does gray death treatment ever begin?

The answer is Yes, what's Ambrosia is for then? If we ask differently, did JC had any (chance for) treatment?

There is exactly one vial in the game - in a chemical lab on upper decks of PRCS Wall Cloud. You can deliver it to Stanton Dowd to help him with his sickness. It has no effect if you use it on yourself, other than being rewarded with the message, "Strange...nothing happens..." http://deusex.wikia.com/wiki/Ambrosia_vial

Except that I dont recall anything. All medics and medical institution offer only HP restore and Augmentation installation.

Solution 2:

JC doesn't need Grey Death treatment. There's actually indications that Grey Death isn't just a disease but also a way for certain people to tolerate nanotech augmentations. For example, when you find Paul in the MJ12 lab a datacube nearby says Paul tests positive for Grey Death but curiously does not develop symptoms. There's some other hints that truly curing Grey Death (Ambrosia is not a cure but a temporary treatment) would also provide the key to letting anyone use nano augmentations.

I could swear someone actually tells JC he doesn't need treatment for Grey Death because "...you could say you are Grey Death". That might be cut dialogue that the GMDX mod restores in the game.