"No" is a complete sentence

Technically, "No." is not a complete sentence. A complete sentence must express a complete thought. A subject, (tells who or what the sentence is about), and a predicate, (tells what the subject does or what state of being it is in), must be in there somewhere to express a complete thought.
"Go." is a complete sentence because the subject is implied, ('you', as in "You go."). 'You' is the implied subject and 'go' is the imperative action (telling what the subject does in this case).

"No" is understood as a complete thought, because it is usually used in response to a question, but it really isn't one. There is no subject or predicate, forget whether they are implied or not.

The four types of sentences are:
declarative: makes a statement, ends with a period (or 'full stop'). I love rolls.
imperative: gives a command, ends with a period/full stop. Pass the rolls.
interrogative: asks a question, ends with a question mark. Where are the damn rolls?
exclamatory: expresses great emotion, ends with an exclamation point. Omigod, these are the best rolls ever!

This useless trivia (unless you make it on Jeopardy some day) was brought to you by one who survived fifth grade English.

No. is really just a shorter version of the complete sentence "It is not so." Just as in "Go." the subject is implied, in "No." the verb and subject are implied. And you cannot "forget whether they are implied or not," because it is essential to the function of the word that they are implied.

If "No" is not a complete sentence (or even a complete clause, because a clause needs a verb), then "Yes" is not a complete clause either. Now, use "Yes" as part of a complete clause.

You can't, can you.

The word Yes is always found as a clause on its own, so it must imply a verb.

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therevengingo: In your sentence, the answer is an instance of the clause "Yes". So it still depends on the fact that Yes on its own is a complete clause.

Of course, the sentence "No!" may be used to signify:

• Refusal to carry out a request or order. This may seem to imply a verb; but viewing it as a proposition of the form 'it is not the case that I will do the washing up,' for example, misses the point. If we undertake this kind of analysis, how do we know we should not render it with a different verb, viz.: 'it is not the case that I will follow your order to do the washing up,' or 'it is the case that I refuse to follow your order ... ?'

  The fact that the 'implied verb' is indeterminate argues that it's some kind of philosopher's or linguist's fiction, I think. (This kind of confusion often happens when we conceive language as merely a mechanism for the transfer of information.)

• General horror at a situation. If we take 'no' to be shorthand for some proposition, such as 'I refuse to believe it.' or 'I don't like it at all' we are again forced to forge some definite propositional sense from a sentence which may be more or less 'non-cognitive' (as proponents of the boo-hurrah theory of ethics, such as A.J. Ayer, might say.)

• A response to another's actions, rather than to their words; a negative imperative. And in this case it seems easy to see that the sentence may well require, or admit of, no further explication, and still be perfectly clear in its meaning.

• Disagreement on general, but unspecified, grounds. Examples are left as an exercise for the reader.

That it is a sentence can be seen from this: it might be the only word ever spoken to us by someone, even someone that we had never ourselves spoken to -- and yet, given the right circumstances, we would understand them perfectly.

This is a reasonable test of sentencehood: whether, isolated like this, it can be understood. Often this is a case of chasing dangling parts of speech, as in the grammar-school idea of a sentence given below by Phinslit, but this task is subservient to the idea of semantic completeness. Sometimes, understanding can occur without the kind of specificity required by the grammarians.

Regards the corresponding question of whether 'Yes' must always imply some propositional scaffolding in order to make sense, a close reading of the final chapter of Ulysses is advised.

This brings up an interesting topic. Before I go into the nitty gritty, I'd like to mention that in Latin, there is no direct way of saying "yes" or even "no." Typically, if one Roman were talking to another and said "Are you the farmer who lives in a province of Asia?" the other would respond either "I am..." or "I am not...." Anyway, without further ado, the reason you're -really- reading this node: Whether "no" is a complete sentence is dependant enirely on the definition of a sentence. Do me a favor, friend, head on over to sentence and scroll down to Webster's entry. Lots of entries, eh? For the moment, let's skip down to the entry under grammar.

"A combination of words which is complete as expressing a thought, and in writing is marked at the close by a period, or full point."

Ok, that's interesting. A sentence requires a few things: some words that make sense, and that little dot thing. Okay, but what on earth makes a bunch of words "complete as expressing a thought"? What makes a sentence complete wih respect to expressing a thought? Well, if we scroll down Webster's writeup a teeny bit more, we find he says

"Sentences are simple or compound. A simple sentence consists of one subject and one finite verb; as, "The Lord reigns."

Ahh, so there we are! A subject and a verb. Is no a subject and a verb? No. But wait, isn't that a complete sentence? I just expressed a complete thought and put that little dot thing after it! Unfortunately, grammatically speaking, it is not a complete sentence, simply because it lacks a subject and a verb. You did express a thought, but not by means of a subject and a verb (and a dot).

To refute FordPrefect's saying that "no." as a sentence has an implied verb and subject, I'll say this: In English, we imply subjects all the time (Go to the store. Who? You, go to the store). It's how we use the imperative mood; we are giving a command, and the one whom we are addressing understands that s/he is the implied subject of the command. This is a very special case and we rarely imply subjects outside of the imperative sense. Implying a verb is something we don't do. The word no does not imply a subject and a verb all on its own, grammatically speaking. Of course we understand that, if a mother tells her son to go to his room and the son says no, the son really means "no, I shall not go to my room."; however, because we understand what he means does not mean that the word no implies the subject and verb in a sentence.

The issue here is not so much what "no" means--we all know what it means--but rather, with what a sentence denotes. If this node were titled " 'no is a complete thought' " we wouldn't be having this discussion. A sentence contains a complete thought expressed by a subject, a verb, and a period. No is neither a subject nor a verb; thus, it can not be a complete sentence.