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)