Yes, “no” is separate from the clause containing the verb. No, these are not double negatives.
Whether you consider “yes” and “no” clauses on their own is more contested, but it really doesn’t matter—they are not part of the clause that the double negatives would appear in. The fact that they (“yes” and “no”) are normally followed by a comma—and can perfectly well be followed by a period—indicates that they are separate entities:
No, it is not.
No. It is not.
In cases of double negations, such separations are not possible:
I ain’t got nothing against double negations.
*I ain’t got, nothing against double negations.
*I ain’t got. Nothing against double negations.
The last two of these are quite obviously not valid, since “I ain’t got” (or “I haven’t got”) is not a complete sentence.