There are two things called his incompleteness theorems, but only one actually deals with completeness. The other deals with consistency, which is not coming up with contradictory answers. Basically, it says that any system that asserts it is consistent is necessarily inconsistent. This is I suppose related to things which are true about the system which cannot be proven true in the system, but only in a roundabout way.
Computability and completeness are also definitely related, but, not quite so tightly as you implied. Computability is a property of algorithms, while completeness is a property of axiomatic systems. Algorithms operate on axiomatic systems, but are not the systems they implement, and there's no inherent connection between being able to prove statements about a system, and the algorithms used to compute proofs for that system. As I said, SOL can prove FOL to be complete and consistent, and you can do this on a universal turing machine, but FOL cannot prove itself to be complete and consistent even doing this on a universal turing machine.
Of course, if your computer _is_ an axiomatic system (and all are), then the incompleteness theorem(s) do say things about what that computer can do, and we know that all turing complete computers have precisely the same sorts of constraints in that regard.
The real thing that was irksome, tho, was the assertion that there are some things that are obvious to humans but which are uncomputable. This is just unfounded. Unless you're supposing human brains are somehow not computing things (collections of neurons are computers, after all, just not von Neumann architecture), there is no way for this statement to be true.
Computability and completeness are also definitely related, but, not quite so tightly as you implied. Computability is a property of algorithms, while completeness is a property of axiomatic systems. Algorithms operate on axiomatic systems, but are not the systems they implement, and there's no inherent connection between being able to prove statements about a system, and the algorithms used to compute proofs for that system. As I said, SOL can prove FOL to be complete and consistent, and you can do this on a universal turing machine, but FOL cannot prove itself to be complete and consistent even doing this on a universal turing machine.
Of course, if your computer _is_ an axiomatic system (and all are), then the incompleteness theorem(s) do say things about what that computer can do, and we know that all turing complete computers have precisely the same sorts of constraints in that regard.
The real thing that was irksome, tho, was the assertion that there are some things that are obvious to humans but which are uncomputable. This is just unfounded. Unless you're supposing human brains are somehow not computing things (collections of neurons are computers, after all, just not von Neumann architecture), there is no way for this statement to be true.