I will re-phrase:
Except as noted here, Fast Healing is just like Natural Healing.
Except as noted includes "A creature with the fast healing special quality regains hit points at an exceptionally fast rate, usually 1 or more hit points per round, as given in the creature’s entry"The Dying condition effects Natural Healing, and to a lesser extent, Fast Healing, but not the "except as noted" points of Fast Healing, and therefore would not interfere with the previous quote. The previous quote would then unarguably still apply and act as normal, which would include stabilization.
Better?
Except that the quote given is not the most recent and threfore official rules.
The text in the Rules Compendium (pg 74) does not include
"At the beginning of each of the creature’s turns, it heals a certain number of hit points (defined in its description)."
Which is the text that allows it to work for stabalization.
The text under "Recovering without help" also in the Rules Compendium (pg 74)
"A creature that becomes stable on its own (by making the 10% roll while dying) and that has no one to tend toit still loses hit points, just at a slower rate. It has a 10% chance each hour of becoming conscious. Each time it fails this hourly roll to become conscious, it loses 1 hit point and doesn't recover hit points through natural healing.
Once it becomes conscious and disable, an unaided creature still doesn't heal naturally. Instead, each day it has a 10% chance to start healing naturally (starting with the day it becomes consious). Otherwise, it loses 1 hit point. Once an unaided creature starts recovering hit points naturally, it's no longer in danger of losing hit points in this way."
So it mnakes stabalization roll (10%) first day to prevent losing a hit points on an hourly basisis until it makes it.
Then it makes another 10% roll each day - starting with that day to see if it can heal naturally.
So no "auto stablize" here for fast healing, still requires the 2 10% rolls before it "kicks in"..