One has to be careful how the words Noun and Verb are understood, if one wants a good answer. Semanticists talk about Entities and Events, and leave Noun and Verb as formal categories, dependent on criteria of usage (can it be a subject? can it take a definite article?) instead of meaning.
Frawley (Linguistic Semantics, 1992 Ch.3 "Entities", p.62) notes that,
- while it is certainly not true that nouns always refer to persons, places, or things,
- it certainly is true that persons, places, and things are overwhelmingly expressed using nouns.
This is because nouns normally refer to entities, and verbs normally refer to events (including states).
In other words, those definitions of noun and verb from grade school are completely backwards.
There's all kinds of linguistic evidence that most languages do distinguish between noun and verb.
But that's not evidence that categories like
Verb can be attached to individual words.
At least not permanently; lexical categorization is a separate issue. Pretty much any English word, for instance, can be used as just about any open-category part of speech: noun, verb, adjective, adverb. But we have good tests for what a given word is being used as -- in a given sentence. Not generally.
However, the joker in the question is that it presupposes that there are words in a language.
Most languages do have words -- i.e, they can put word-level constituents together into larger ones.
But there are polysynthetic languages, like Eskimoan languages and Salishan languages. In a polysynthetic language there is usually a very simple root system with dozens of derivations and inflections that get added on to form even the simplest utterance. And these roots are very general, and have lots of metaphoric and idiomatic associations -- like any other language -- so their possible forms get big very fast. Both in the sense of getting longer and longer, and in the sense of there being almost infinite extensibility of words.
It says something about Skagit (Puget Salish; Northern Lushootseed) that over 3/4 of the nouns in the language start with the overt nominalizer s- --even the words for 'man' and 'woman'. They're built on CVC roots, as is most of the rest of the language. So in Skagit, while there are nouny things and verby things and you can tell the difference in a given sentence, they don't split up the descriptive labor the way we're used to, and the roots are normally neither verby nor nouny, but can be made to refer to either. Plus there isn't a really well-defined boundary between words and sentences.
So serious scholars will question the presuppositions.