It is rather a case of the use of a partitive article vs a definite article

Je ne mange pas de viande

means you never eat any meat of any kind, because of religious reasons, because of dietary restrictions, because you are a vegan, etc. It is also OK to write je ne peux pas manger de viande (it stresses then it is a rule you follow)

Je ne peux pas manger la viande

means you are not willing or not able to eat a specific piece of meat that is present here and there while you are talking. Eg, you are not willing to eat this piece of meat that is on the table because you are a vegan, or maybe because you feel this piece is not fresh or has a bad taste, or because you are on diet right now, or you cannot eat it because it is too hard to chew on. It does not necessarily mean you never eat any meat at all.

You can find the same meaning in the affirmative form, so it is no linked to the negation.

Je mange de la viande

is a general statement: you eat meat, of any kind, at any time

Je mange la viande

refers to the piece of meat you are eating right here, right now

Note that you can also find the same la for a general statement if you specify which kind of meat you refer to.

Je ne mange pas la viande qui vient de ce boucher (parce que je soupçonne qu’il ne respecte pas les règles d’hygiène)

In addition to Greg’s answer, I would risk some shaky hypothesis in order to explain what la in pas de la modifies in terms of logic.

If you’re talking about the particular meat on the table, you’re using la, article défini: it’s a particular proposition. If you’re talking about meat in general, as a concept, you’re negating an universal proposition, but je ne mange pas de la viande is the denial of a particular, not of an universal (it would be: je ne mange pas de viande [du tout]). Any use of a particular article would produce a particular proposition.

Thus, pas de viande expresses an absolute negation (aucune viande). Pas de la viande would be partitive (no part of this meat). By the way, it seems to me that * pas de * is synonymous here with * aucun*, whose TLF (art. “aucun”), following Grevisse, tells us that

Le subst[antif] est introd[uit] à l’aide de la prép[osition] de. Toutefois, si ce nom
est déterminé par un compl[ément] ou par une prop[osition] rel[ative] ou, plus gén[énéralement], si l’on exprime vraiment l’idée partitive, il demande du, de la, de l’,
des (Grev. 1964, § 329)

As we do not want to express a partition, but a zero quantity, no definite article.