That is the origin of the infamous "dark matter" hypothesis.
There are two main attempts to explain why galaxies spin faster than expected based on their apparent mass.
1) There is more matter in them than is "apparent". That is, there is matter which isn't apparent, i.e. it's "dark".
2) Gravity doesn't really follow the rule GMm/R^2. Perhaps, at distances much greater than the size of our solar system, it follows a different rule. Or it follows the same rule everywhere, but it's not GMm/R^2, but it's something that can be approximated as GMm/R^2 at scales up to the size of our solar system.
All things being equal, I favor #2, because I don't think you should make up something you haven't observed just to avoid fixing a broken theory. Dark matter could be like the "luminiferous ether" imagined before Einstein's special relativity.
However other observations seem to back up the dark matter theory, for example observations of the way galaxies collide seem to show the dark matter portions having a distinct effect.
It's supposed that maybe dark matter is some kind of matter that doesn't interact with any of the forces of the universe other than gravity. It's not too hard to imagine, really. We know of forms of matter that don't interact with the electromagnetic forces, such as neutrons. Why shouldn't their be forms of matter insensitive to the other forces as well?