The 1954 Godzilla could not. I personally don't think its regeneration was that good, so having a limb severed would have meant permanent damage.
The Showa Godzilla might be able to, but I have my doubts about that as well. We've seen the kind of damage Showa Godzilla took, but it's still debatable. Thinking back on the effect Hedorah's body fluids had on Godzilla, Godzilla's skin was totally gone, and all that was left was a skeletal hand, and Godzilla later loses an eye in a similar manner. Now, later on, Godzilla's regenerated these affected areas and everything is pretty much back to normal. Obviously the skin tissue has successfully regenerated, and it appears that Showa Godzilla can repair nerves, blood vessels, et cetera. Now, the eye is what intrigues me. I'm not sure if other animals/reptiles/amphibians can regenerate an eye, but it appears Showa Godzilla can. This may be an indication that this incarnation is capable of regenerating a limb, but I can't say that with any certainty. Godzilla has an eye again, but we have no way of knowing if its vision is fully restored or if there is any lingering damage. If at all possible, Showa Godzilla could possibly, over a certain period of time, regenerate a limb, but I have to question have successful that regeneration would be.
For Heisei Godzilla, I'm inclined to believe that it may be possible, considering the sort of damge it takes on. Biollante's vines shooting through Godzilla hand and shoulder most definitely shattered bones, but Godzilla shrugs it off and keeps on fighting. It has rapid regeneration, and apparently much more evolved from its Showa counterpart. The only doubt I have comes from when Mechagodzilla succeeded in rupturing Godzilla's "second brain", only now we know that area would be a bundle of nerves and not a brain. At any rate, going by the "second brain" theory, Heisei Godzilla apparently has limits to its regenerative abilities. In other words, it cannot regenerate any organ that cannot be regerated. Nerves are complex, but, for the most part, they can regerate and heal, unless there's been massive burn damage or something. Brain cells and liver cells, however, do not repair themselves. Godzilla's inability to heal its "second brain" is mostly likely because of this. If Godzilla's liver were to rupture, chances are that it could not repair that, either. If not for Rodan's intervention, Godzilla would have remained paralyzed. Even if it survived through Mechagodzilla's arsenal (which would mostly likely happen, at least for a time), Godzilla certainly wasn't going anywhere. Eventually, Godzilla would have most likely died. So, while Heisei Godzilla can regerate virtually anywhere on its body, chances are that organs that do not naturally regenerate would be just as vulnerable to the monster. What does that mean for limb regeneration? Owing to the way Heisei Godzilla regenerated after taking attacks like what Biollante dished out, chances are that it may be possible. However, there is still the question of how good that regenerated limb would be.
Now, for the Millennium Godzillas, this is going to be a bit tricky:
G2K Godzilla: In the movie, Shinoda and Miyasaka seem to hint at the possibility that Godzilla's regeneration is probably that great, but I really have my reserves about that. From what I observed in the film, G2K Godzilla can regenerate skin tissue easily, but I somehow doubt that a limb could be regenerate and, if it could, it would take a much longer time to do that.
GxM Godzilla: It may be possible, but I deem it highly unlikely. This Godzilla incarnation didn't get dealt that much in the way of severe damage, save for Megaguirus's stinger, so I doubt it could regenerate a limb.
GMK Godzilla: Here, I would have to agree with the majority that it's likely to be possible in this instance, owing to spiritual connections of this incarnation. Considering the fact that Godzilla's dismembered heart is beating strongly at the end of the film, I can see this one regerating a limb, albeit most likely covered in loads of scar tissue, after a long period of time.
Kiryu Godzilla: To be honest, I have my doubts about this incarnation as well. After GxMG, Godzilla returns with a scar on its chest from the AZC, which indicates to me that it has limited or weak regenerative properties. Additionally, it didn't seem as though Godzilla was faring well at the end of TSOS, and the wound it received was pretty bad, though something that Heisei Godzilla probably would have recovered from rather well. So, for this incarnation at least, I doubt very much it could regenerate a limb properly, if at all.
GFW Godzilla: Owing to the amount of fights Godzilla got into, a person would think that this Godzilla would be okay but, once again, I have my doubts. Godzilla was portrayed as a far more powerful being in GFW, and therefore received what we could say is smaller amounts of damage. GFW Godzilla has speed, power, and whatnot on its side, but it didn't get much damage save for what Gigan and Monster X and later Kaiser Ghidorah dealt it. GFW Godzilla does have some regeneration, but I don't think it could regenerate a limb.
There are as many people as there are stars...
There are as many encounters as there are stars...
-Ruri Hoshino, Nadesico: the Prince of Darkness