Logan R. Kearsley has written a comprehensive answer on one angle. I will throw a hint on another angle: if you have enough Noun Incorporation (linguistics) and polysynthesis at a language, you’re going to end up with languages where what European languages treat as nouns or adjectives usually end up as affixes—so what look like words are mostly verbs. In fact, from time to time you do hear people claiming that some such languages (almost always Amerindian languages) don’t have nouns, though my recollection is that the claim is marginal.
So you can in principle have languages with just verbs. However in practice you do have affixes that will tend to signify arguments rather than predicates—so you’ve really just pushed the noun/verb distinction down into morphemes rather than words.
Oh, I see Logan has also written on the converse, whether a language can have just nouns: Is it possible to make a language with just nouns and adjectives?