[Guderian93]

The best answer I could find is that the element's d-band electronic structure needs to be filled for it to be a precious metal and this results in its stability.  Can someone clarify as to what this means?

[Enthalpy]

I don't see a relation with a full d-band in the joined table. Many remarkable exceptions in both directions.

[Corribus]

Your question is a bit of a tautology.  Why don't precious metals oxidize or corrode? Well, it is that very property that defines them as "precious"!  It's a bit like asking why breakfast isn't eaten in the evening.

[Guderian93]

It really isn't tautology.  I'm asking why a group of elements that includes platinum, rhodium, gold, iridium, osmium, palladium, rhenium, ruthenium, germanium, beryllium, silver, gallium, indium, tellurium, mercury, and bismuth do not corrode or oxidize easily.  This just means they don't react easily with stuff like hydrochloric acid or oxygen in the air.  Why, what about their electronic structure makes them more stable than say, iron or nickel.  That is my question because I can't seem to find a straight answer anywhere.

[AWK]

This is not a good point of viev. Many of this elements reacts easily in so called specific conditions, eg. Ag and Hg react with S and even H₂S at the presence of O₂; Pd reacts with acetic and other organic acids  in air.
You did not mention Al which is stable in air and in even concentrated HNO₃ (diluted HNO₃  readily dissolves Hg and Ag) but a drop of of solution of any Hg salt causes  a very fast corrosion of Al.
Frankly saing only Rh is totally precious - even aqua regia dissolves it very slowly.
Some elements you mentioned are non-metals and in any case cannot be treated as precious metals.

[Enthalpy]

[...] why a group of elements that includes platinum, rhodium, gold, iridium, osmium, palladium, rhenium, ruthenium, germanium, beryllium, silver, gallium, indium, tellurium, mercury, and bismuth do not corrode or oxidize easily. [...]

The list is a bit heterogenous:
- Do really all these elements resist oxidation?
- The elements cited first are less oxidable than hydrogen and are the ones usually called "precious"
- Other elements may resist oxidation under the proper conditions because their oxide layer adheres well and is tight to oxygen. Al, Si, Ta, Nb, Ti, Cr, Ge and few more. Usually not called "precious".

The category protected by the native oxide layer is very sensitive to alloying elements, metallurgical conditions, and badly known factors. Little Ga alloying Al ruins it, as do Cu or Zn. >13%Cr in Fe protect it, but not against Cl^- in water, and if such a stainless steel must be welded, we specify <0.02%C in it so the seam resists corrosion.

Against corrosion worse than seawater or fruit juice (or some simple acids), the native oxide layer hits limits. It's also an obstacle to electric contacts, brazing... In which cases the choice reverts to precious metals.