Tom,
It goes beyond "conjecture", as if the "conjecture" of dark stuff was the same as the "conjecture" of god's hand.
The point is, given known (that is, well verified by many, many experiments) physical "laws", such as the law of gravity, there must be more matter--much more matter--out there than we can see given observed facts, such as that the galaxies have not spun apart.
In this sense, dark matter is inferred from observation. You could call it the hand of god if you want. But the problem is that doing so does not advance our understanding of the real world, or our ability to manipulate it, unless we can come up with a "law of the hand of god".
Falsifiability is much talked about, and Popper much refered to (not, thank goodness, by you) in such debates. But while it is important that physical laws be falsifiable, at least in principle, it is a bit of a red herring and, as is now very well known, it is not the principle of falsifiability that in fact drives scientific enquiry.
And it is not a good basis on which to attack religion either. All the religious need do is generate in principle falsifiable "laws of god" and say, there you are. Of course, some would, in fact be falsified. But this would not logically entail the end of religion, any more than falsification of, say, Newton's law of gravity entailed the end of religion. Rather, new physical laws were devised that accounted for the falsification.
At the end of the day, the new physical laws prevail because, in some instances at least, they are more useful than the old ones. They make correct predictions about things we care about, whereas under some circumstances (none that are likely to much affect our day-to-day lives) the old laws make wrong predictions.