Rose is a human, while River is a greatly mutated and modified one.
There is literally nothing in Rose's body that could make use of regeneration energy. Her biology is not that of a Time Lord and has no method of processing regeneration energy whatsoever. To expect regeneration energy to have an effect on a being without some semblance of Time Lord physiology would be like expecting anything without a semblance of Weeping Angel physiology to be quantum-locked when observed, which obviously does not happen.
River Song, unlike Rose, has been mutated via her prenatal exposure to the Time Vortex (which was noted as being a downscaled version of what caused the Time Lords themselves to evolve into what we know them as) and subsequent genetic modification by the Kovarian branch of the Silence. River clearly has numerous physiological differences from regular humans; for one, she can regenerate and therefore must be able to physically use the energy, for two, she is time-sensitive and her brain can cope with complex temporal mechanics, and for three, she has an extra strand of DNA, as seen in "A Good Man Goes to War". The episode "Let's Kill Hitler" also demonstrated that she could use her own regeneration energy in order to heal the Doctor.
So all in all, River is far more "Time Lord" than she appears to be, and thus, it stands to reason that regeneration energy can only heal beings whose bodies naturally use regeneration energy.
More importantly, however, is the fact that the damage was already done to River's wrist and was not a very significant injury in comparison to the whole of the Time Vortex actively destroying Rose's body.
I would expect that whatever regeneration energy the Doctor had would not have done any good, especially since the issue wasn't that she needed healing, but rather, than she needed the Vortex to be extracted from her before it killed her.