Suppose that we are not talking about any current dictionaries but one a bit more complete, with more examples of usage and meaning, something more precise about the context each word can be used in, something with phrases and idioms with specific meanings. This isn't terribly idealistic, just probably currently out of reach because of resources and desire.
First presume that we are given such a dictionary as a foreigner (the dictionary is written in our foreign language to help us know how to understand English). Then with the assumptions above I think it can be said that this augmented dictionary is the English language for the foreigner (yes, there are difficulties with translation, but not impossibilities and I'll presume that the assumptions above take care of these difficulties).
But instead, let's presume that it is self-defining, that the dictionary is in English, it is a definition of English in English. I don't see how this could be a definition of the language, because an understanding of English must be assumed to be able to interpret the definitions. There might be room for the 'cryptogram' idea, in the sense that the foreigner might be able to extract meaning out of the English definitions recursively and somehow extract meaning out of the circularity.
Those are just supposing we can solve the limitations of a dictionary.
But then also, even if individual words have every nuance explained, those words (or even phrases or idioms) are not the entirety of language. There are all the extra-lexical parts of a language: syntax and semantics.
And then there are all the extra-linguistic things, those not part of the spoken utterances but that surround them, the culture, the day-to-day interactions, and the economic and material climate, that restrict what is appropriate to say and what not. There's the anecdote about the professor who visits a country to give a highly technical lecture in full fluency, but can't even open his mouth to order in a cafe or talk to a child.
A dictionary is certainly a necessary component of a process to 'reconstruct' a language. And one could communicate passably in a language just spouting a set of words picked from a dictionary. But there is a lot more needed to define a language.