No they don't. And there is nothing 'fundamental' to these 12 tones.
To clarify:
I meant "fundamental harmonic" in the sense of the lowest frequency in
a harmonic series, not in the sense of being fundamental to anything
more profound. Each note does, in fact, have a fundamental harmonic,
but it seems you may be right in the sense of there being nothing
fundamentally important about notes in the making of music writ large.