Consider the sentence "I would like to say that he is incorrect."
I don't know if the correct translation of this sentence is « Je voudrais dire qu'il est incorrect. » or « Je voudrais dire qu' il soit incorrect. »
I want to say the correct translation is the second one because I'm inclined to say Je voudrais dire is not an independent clause but I'm not so sure.
As far as I understand, the sentence I would like to say has two different meanings.
It might just tell that you are about to say something that you feel is important, like in a speech that starts with:
First, I would like to say thank you...
In that case, a translation might be:
En premier lieu, je tiens à (vous) dire merci...
The second meaning occurs when something is preventing you to say it.
I would like to say the truth, but I'm not sure it's a good idea.
You can translate that by:
Je voudrais dire la vérité, mais je ne suis pas sûr que ce soit une bonne idée.
This second form might also be used when the first meaning is intended:
En premier lieu, je voudrais vous dire merci...
but the opposite is not true. Je tiens is much less likely to be used with the second meaning.
Depending on the expected meaning, your original sentence might then be translated either by:
Je tiens à dire qu'il a tort1.
or
Je voudrais dire qu'il a tort.
Finally, je voudrais dire qu'il soit incorrect is not grammatical. There is no place for hypothesis or uncertainty after je voudrais dire.
1 We wouldn't say qu'il est incorrect or qu'il n'est pas correct because that is more likely to mean that he is "rude, inappropriate".*
In short : Je dirais bien qu'il a tort
The doubt being on the fact that you will say... or not. That you will judge... or not.
Not about the fact that he his wrong. =>
Principale au conditionnel et subordonnée à l'indicatif.
Of course, things would have been different if you had wished to say :
It could be possible that he is wrong or I'd say that he might be wrong
No, "Je voudrais dire" is not an independent clause, it's what you call "une proposition principale", that is "a main clause", in particular, a clause, which unlike an independent clause, cannot always stand by itself; often you have to add a subordinate clause so as to have a complete meaning. Nevertheless, that has nothing to do in deciding which mood should be used. You can't say "Je voudrais dire qu' il soit incorrect.", in any case. That leaves the first sentence as a correct translation.
You have to use the subjunctive when the verb is simply "vouloir"; that makes the reason of the subjunctive when you add an infinitive to "vouloir" somewhat puzzling;
Je voudrais qu'il soit correct.
La mère voudrais que sa fille guérisse.
The reason is not so remote from a straightforward understanding though: when no infinitive is used the immaterial nature of the action induced by "vouloir" is conferred to the subordinate; when the infinitive is added, it is this verbal form that does not confer the idea of an action that has been realised.
