I started to write this, and my computer froze and gave me a terrible time about coming back on reboot. The video still isn't right--the colors are a bit off--but at least it's running again.
Good answer from everyone; I would go with Scott's second answer. Follow my reasoning.
We're assuming that the verser's heart was removed, and that this counts as damage, but that he was connected to life support and fully recovered in that condition. Then the life support was disconnected or shut down, and he died and versed out.
"Stable form" means he is fully recovered and alive in that condition. For that to be the case, he must have all of his vital organs, or suitable replacements for them. The life support equipment must in this case be those replacements, and therefore are now part of his body. Just as with Michael di Vars' robotic arm or Joseph Kondor's Meritronics Cybereye, these external life support machines are functionally part of his body.
Further, if you were beheaded your head would go with you and be replaced upon arrival; if your heart was ripped from your chest, it would be restored and reconnected. Thus, when the doctor kills you by disconnecting or shutting down the life support equipment, you verse out, but you take your new "body parts"--the life support equipment--with you.
You might have to find a power supply for it quickly. However, I can't imagine that anyone would build such a device without an internal backup power supply hard-wired into it, so that will extend the life--particularly since it makes sense to me for that power supply to be fully charged as part of the stable form healing process. Further, I think it would recharge itself each time the character died and returned to life, so you'd probably have at least half an hour, maybe twenty-four hours, to find a power supply or another solution to the problem. Of course, our mad doctor might have disconnected and removed the internal power supply--but that might also heal itself under these circumstances.
As an aside, Bob Thomas is not in a magical space. He is in a dimensionally engineered briefcase, on the order of the interior of a TARDIS. It is important that this is an entirely technological device for several reasons. The obvious reason is that Peter Adams, who owns the briefcase, intensely disbelieves magic completely, and could not operate a magic device not consider such a thing possible. The more important reason is that being a technological device it would continue to function in its last state after versing out until someone attempted to operate it, and therefore the dimensional space would remain intact if no one opened it in a low tech world. In a low mag world, such a space would become an instant botch, as an always active mag device is a disaster waiting for a chance to happen.
--M. J. Young