grammar - What do you mean that it's wrong?

I wonder whether different examples can shed some light: 
 

How can you say [that] it's wrong?

There doesn't seem to be anything odd about the nominative subordinate clause "[that] it's wrong".  To my eye, it looks like an direct object.  It's the thing that can be said.

The part that seems odd is that "how can you" and "why would you" express the same sentiment in this context.  That doesn't hold true in other contexts.  Questions like "how can you eat so much?" and "why would you eat so much?" expect different answers, such as a high metabolism and a low self-esteem respectively. 

However, that's a question of modality which we can ignore for the moment. 

What do you mean that it's wrong?
Why do you say that it's wrong?

The subordinate clause still appears to be a direct object.  These two questions seem to express similar sentiments and expect much the same range of answers.  Both questions allow the original statement to be supported or explained, or for the implications of that statement to stand as an answer. 
 

The oddity here is that the "what" in that question acts like "how" and "why" usually behave.  "How" and "why" are adverbial interrogatives.  They can be parsed as adjuncts rather than arguments.  Ordinarily, "what" is a pronominal interrogative, which isn't a suitable adjunct on its own. 

We can explain this oddity if we assume an elision:

What do you mean [by saying] that it's wrong.

Here, "that it's wrong" is the direct object of the gerund "saying".  The entire prepositional phrase "by saying that it's wrong" is an adjunct to the verb "do mean", while "what" acts as its direct object. 

If we do not assume the elision, the next obvious possibility is that "to mean" allows "what" to act as an adjunct.  The questions "what do you mean that it's wrong?", "how do you mean that it's wrong" and "why do you mean that it's wrong" expect similar ranges of responses, even though we've progressed from the utterly unsurprising to the highly questionable. 

The elision seems easier to support. 

How do you mean that it's wrong? 
You mean that it's wrong, but how? 

Given a clear adjunct, we can separate the question that it asks from the statement that it modifies.  The same doesn't hold for "what": 

What do you mean that it's wrong? 
*You mean that it's wrong, but what? 

Once the verb "mean" has an obvious direct object, the word "what" no longer makes sense.  It doesn't act like an adjunct from other positions, even though we haven't changed the governing verb. 

What do you mean that it's wrong? 
You said that it's wrong, but what do you mean? 

This transformation practically begs for the restoration of at least one elided word.