The fact that someone might make a mistake in doing something does not show it cannot be done. More generally, You seem to be mistaking validating a program's source with the question of whether it performs its intended purpose. These are different things, and attempting to conflate them will only lead to confusion.
Then your mistake appears to be in failing to see that your perceived edge case does not invalidate the first sentence of my reply. If the sole purpose of a parser is to syntactically validate its own source (which is not the case for a compiler's parser, by the way, not even if we expand 'its own source' to 'arbitrary input'), then if it does that correctly, that's all there is to it.
It is a general rule that those who avoid answering a question do not have an answer, and this is no exception. Here, You completely misunderstand Chomsky’s hierarchy: By your inverted-hierarchy argument, the simplest regular language would be complex enough that incompleteness would be an issue in its validation.
A self-hosted compiler is definitely not a case of the liars paradox, so I don’t think this applies. In fact, it doesn’t matter what the parser is written in, at some point you do have to trust that it actually follows the spec, probably through testing.
In the case at hand, correctness of the validator expression V clearly means "V determines well-formedness of any regular expressions" which is clearly not implied by "V is well-formed" (a much weaker statement because ".*" is well-formed but matches everything). Therefore, when applying V to itself, we only learn if a weak requirement for V's correctness holds.
Similar, perhaps, to validating whether a given number is a Gödel number of a well-formed logical statement rather than assessing the verity of the logical statement it encodes.
Also, I am not saying whether it is or is not possible to build such a regular expression. Rather, the question just doesn't tick the boxes of either, the Liar's Paradox, nor Gödel's Incompleteness result -- contrary to what was suggested. So you could still be right but for different reasons.
It's not possible to do this for regular expressions, but it is possible to do it for context free grammars. You can write a context-free grammar in Backus-Naur form that recognizes all context-free grammars in Backus-Naur form:
