It's true that his healing power can repair his body, but it takes time (based on the movie series). Hot adamantium (which destroyed his skeleton to take its space) was able to destroy his body internally faster than the healing process. But, still his body wasn't destroyed after surgery. How?
The Short Answer: The reason hot Adamantium did not destroy his body was that it could not be applied "hot" because heat cannot affect Adamantium at all. The process they used was a cool one and once the Adamantium stabilized, it could not be affected again, without the application of significant technological, advanced mutant or cosmic-level forces.
The Longer Answer: Wolverine's mutant ability: Hyper-regeneration
Wolverine (no matter which continuity you view him in) is a mutant with the power of hyper-accelerated regeneration. This is a regeneration ability nearly on par with the Hulk in terms of the ability to regenerate damaged tissues. The cellular aspects are not clear but there is the standard canard (meaning pseudo-science) used by Marvel's mutants of using mass or energy from another dimension to replace his bodily mass that is lost when he is damaged severely.
With that said, all of Wolverine's abilities are related to his increased tissue regeneration and his body's perfect cellular recall of his tissues, organs and organic systems. He does this without any conscious awareness and appears to be able to do this even if his heart has stopped beating. Which implies his body's energy and matter source to be completely independent of his body!
This is not so strange, considering all of Marvel's energy manipulating mutants also have an energy source outside of their body as well, considering even if they were able to convert 100% of the food's energy consumed in any period, they would not be able to perform even one of their amazing super-feats. Only the Beast manages to maintain somewhere near real world credulity.
Wolverine's Superior Senses
Wolverine's regeneration has led writers to claim it has also given Wolverine superior sensory abilities, stemming from his body's perfect replacement of cellular tissue, however this is not logical such as his sense of smell that rivals that of a bloodhound. For him to have such powerful senses, he would have to have both cellular superiority and neurological enhancement in the sensory region of the brain. So in my mind he possess TWO mutations, one for his regeneration and the other for his increased sensory capacity. Both make sense in terms of evolutionary enhancements so I don't have an issue with that.
Adamantium (not to be confused with Vibranium)
Adamantium, a mysterious metal, likely of extra-terrestrial origin, that boasts the honor of being the hardest, most durable and nearly indestructible metal of the Marvel Universe. A metal so hard and so durable it cannot be affected by any form of energy directly. A metal heavier than lead and mercury and likely to be as poisonous. So how do you manipulate it?
You would have to manipulate it at the molecular or atomic level, reducing its molecular bonds and isolating single atoms of Adamantium able to be re-bonded to other matter. The scientific capability to perform this feat is limited to groups like Advanced Idea Mechanics or super-scientists such as Reed Richards. Cosmic beings such as the Silver Surfer or Omega-level mutants such as Dark Phoenix and Magneto might (and in the case of Magneto, did - he extracted the Adamantium from Wolverine's body with his magnetic powers) be able to directly affect the material at a rudimentary level using their powers.
So how did Wolverine end up having his skeleton bonded to a metal that should be impossible to shape and highly poisonous to a living being? The answer is found in his mutation and the creativity of human scientists in the Marvel Universe.
Marvel writers created a material called Beta Adamantium, that's how. Beta Adamantium appears to be a molecular analog (meaning it looks like adamantium, has properties like Adamantium but is able to be manipulated without access to cosmic powers or mutant magnetic or telekinetic abilities.) While in science, such analogs are rare, in the Marvel Universe, this appears to be one of the few times where the copy is equivalent to the real thing in almost all ways that matter. Beta Adamantium is akin to Pure Adamantium but able to be manipulated using telekinetic (read that as molecular-level) human technologies such as nanotechnology.
Beta Adamantium and Hyper-regeneration (A better covert agent?)
Wolverine was part of the Weapon X program designed to create technologically-enhanced covert ops soldiers. He was chosen for the bonding process to beta Adamantium because his body could withstand the scoring of his bones using nanotechnology and the bonding process as his body regenerated around the molecular matrices of Adamantium being locked into his body, reinforcing his bone structure, making it very nearly indestructible. His joints, sockets and connective tissues were likely augmented as well because otherwise the increased weight of his skeleton would tear his ligaments, tendons and ball joints apart under the stress.
This is where his hyper-regeneration comes in. In addition to allowing his body to effective bond with the beta Adamantium matrix now enshrouding his skeleton and ligatures his regeneration allows him to survive the poisoning nature of the heavy metal now completely integrated in his body and constantly repairing any tissue damage of his internal structures due to this bonding. This bonding of Adamantium to his skeleton has REDUCED Wolverine's regenerative ability because it is constantly having to negate all of the physical damage of being bonded to this dangerous but durable heavy metal.
During the times he was without his Adamantium enhanced skeleton, Wolverine noted exactly how much better his ability was when he did not have to contend with the bonding. Where bullet wounds took minutes while he was bonded, they healed in split-seconds without the adamantium. This makes his skeleton both a blessing and a curse.
Having it gives him incredible durability and reduces damage to his major organs and spinal column. Not that these injuries would be serious considering, but it would allow him to maintain his level of function as well as giving him access to the superior molecular dissolution powers of his Adamantium claws, he uses to slice through almost any form of matter he comes in contact with. (An article for another day.) Without the endo-skeletal bonding, he loses his matter-slicing claws but gains an instantly regenerating body capable of taking a licking and regenerating in seconds.
It would seem the only real reason he would have had this process done at all, would be to allow him to have access to the Adamantium claws. If the scientists had any real understanding of his powers, they might have considered leaving him without the endo-skeletal bonding, giving him the best of both worlds. Claws that slice through anything and a body that can recover from almost any injury. From their point of view, it probably didn't make sense to have an agent stop to reset his broken bones, even if he could recover from the injury easily. Curiously enough, even when Wolverine was without his endo-skeletal enhancement, we never see him stop to reset broken bones or deal with internal injuries. It was as if his body remembered what it was supposed to look like and restored itself to its template form without him having to stop to do anything like triage.
The situation is unclear.
In the 1993 graphic novel Weapon X, the operation that bonds adamantium to Wolverine's skeleton is shown. He is submerged in a tank while the procedure is performed, and the metal is fed into his body through tubes attached to multiple locations on his body. (The exact locations are unclear; they change from panel to panel. This sequence is mixed with various flashback sequences, and different points in the procedure seem to be shown.)
It's fairly clear that the metal is fed in as a liquid - there's mention of an adamantium feed rate, and Wolverine's body is accepting the metal at a faster rate than the Weapon X physicians expected would be the case.
The story doesn't mention what temperature the metal is at, or even show the metal itself before it's put inside Wolverine's body.
The metal didn't completely replace his skeleton. The same work has a panel showing a medical scan of the bones in his hand and upper arm. It's unclear what's bone and what's metal, or whether there's a consistent outer coating or not, but a caption makes it clear that the adamantium has bonded to the bone, and does not replace it. (A panel from the 1981 Days of Future Past storyline had portrayed Wolverine's skeleton slightly differently. In a possible future, a robotic Sentinel disintegrates Wolverine, and all that is left is his adamantium skeleton - although the "skeleton" looks somewhat mechanical. Going by this image, his entire skeleton is made of adamantium - or at least the bone has a consistent outer layer on it.)