I suspect that it will not be possible to answer this question definitively, due to the contradictory information that Wilkie Collins provides in the text. On the whole, I believe that the weight of evidence points towards carbon monoxide being the toxic agent, rather than carbon dioxide, as I will explain below. Probably the contraditions arise from Collins confusing the terms "carbonic acid" (carbon dioxide), and the very similar term "carbonic oxide" (carbon monoxide), having seen them in Alfred Taylor’s Medical Jurisprudence (1849), available to read on google books.

We are first told that the liquid used to fill the purple bottle is yellow in color:

he [the doctor] took from the specimens a handsome purple flask, high and narrow in form, and closed by a glass stopper. This he filled with the yellow liquid...

While impure acids (such as vinegar, for example) can be colored, medical grades would be clear and colorless. The unusual yellow color would indicate some other substance, such as iodoform as suggested in abligh's answer. The poison gas was created and introduced into the victim's room by means of the fumigation apparatus connected to the room, which consisted of:

a large stone jar, having a glass funnel, and a pipe communicating with the wall, inserted in the cork which closed the mouth of it

The jar would contain some solid or powdered substance, and the mode of use was to pour some fluid into the funnel to react with it. This would release gas, which would escape through the pipe into the room the apparatus was connected to.

As noted by Gareth Rees, carbon dioxide can be conveniently produced by adding acid to a carbonate salt. Carbon dioxide has toxicity threshold of about 30000 parts per million,and so to render the air of a reasonably sized room poisonous would require the use of several kilograms of carbonate ¹ . The amount of acid necessary would depend on its concentration, but would require several litres. This does not seem consistent with the description of the purple flask, which can be held easily in one hand.

Carbon monoxide, however, has a much lower toxicity level of just 200 ppm. It can be produced by dropping iodoform (a yellow liquid) onto silver nitrate. Both of these substances are commonly used in medicine as disinfectants, consistent with the doctor's description of them as "harmless and useful". It is interesting to note that dropping hydrogen peroxide onto silver nitrate liberates oxygen instead. This is relevant, because the use of the apparatus was either as a fumigant or to provide oxygen for patients with difficulties in breathing:

I noiselessly fumigate one of them; I noiselessly oxygenize the other, by means of a simple Apparatus fixed outside in the corner here

So if the Apparatus had been configured to provide oxygen by loading it with silver nitrate, all the murderer would have to do is swap the hydrogen peroxide for iodoform.

Given how strong the case for carbon monoxide is, why can there still be doubt? The reason is that carbon monoxide poisoning is very obvious in the post-mortem, as the blood remains bright red. In the cases of carbonic oxide poisoning provided by Taylor, for example:

the whole surface of the brain [was] intensely red... the substance of the lungs was of a bright red colour

This would be very obvious to any pathologist, and conflicts with the doctor's assertion that "the whole College of Surgeons" would find nothing suspicious about the death. Here I believe that Collins confused the two substances. Taylor's description of the effects of carbon dioxide poisoning are very similar to that given by Collins:

there is nothing very characteristic in the post-mortem appearances...carbonic acid [i.e. carbon dioxide] acts by inducing apoplexy or cerebral congestion

So I believe that Collins intended the poisonous agent to be carbon monoxide, but confused the issue by ascribing to it the post-mortem effects of carbon dioxide.

^{1. Details of the calculation can be provided on reasonable request.}