This question may be inadvertently attempting to understand whether Kryptonians cells derive their strength from innate material strength or as a function of being part of the corporate living being. It's understandable that there would be some confusion because the Superman Mythos has not been readily or easily consistent in its portrayal, which is, in part, because Superman's powers haven't been consistently explained or addressed towards scientifically sophisticated audiences.
We have a matrix of issues that aren't easily reconciled with just one answer without nuance.
If Superman's invulnerability is exclusive to living cells, then how does he keeps his hair, nails, and teeth (non-living keratin or enamel) through every bomb blast, trip through the sun, etc.? However, if all of Superman's bodily parts are simply invulnerable materials... how do we explain them becoming normal when drained of his solar energy (solar flare, solar starvation) or exposed to alternative radiation (red sun, Kryptonite)? An organic being is made of a great variety of materials which can't all fluctuate in strength to such a precise degree based on radiation exposure. he latter would suggest his invulnerability comes from a force field produced by his organs and living cells.
The easiest solution is to say that the force field extends beyond his immediate person, to encompass his hair, nails, and teeth (and costume) despite being non-living.
Yet, we're still left with examples of Superman's hair exhibiting incredible strength even apart from his person. The easiest solution then is to say that these examples are the exception and outliers, not the rule.
In that case, the answer to your initial question is that skin cells which have died and left Superman no longer benefit from the invulnerable force field generated by their living tissue, and are simply subject to normal degradation and decay.