Ok, but I thought we were talking about the transplantation of entire organs. There have been "successful" (as in the patient lived for a while) transplants of baboon hearts and livers into humans over the past 25 or 30 years, but there have not been any transplants of entire pig organs to date.
Doctors really do have high hopes in the future for using pig organs as they are the right shape/size, plus pigs are much easier to raise than primates. It just hasn't happened yet.
Still, it is not correct to say that we are more closely related to a pig than any primate. Relationships among different species is defined by taxonomy - we are in the same order as the primates, different order form pigs, hence we are more closely related to primates.
Sorry to keep flogging a dead horse, but taxonomy is a subject of great interest to me.