There are to many variables to say yes or no. Likely not.  However, the
California warranty may be different than he federal one for emissions
control (if it makes any difference).  The dealer and your relations with
him can also be a major factor.  I just had a Chevy dealer replace two tires
on an 05 that he did not have to, and he managed to get GM to cover them
under warranty.  The tire dealer could not get the tire mfr to cover them
directly.
In any event, the O2 sensor (front or back) can be replaced with Bosch
sensor(s) for less than $100 each (parts, your labor) if the splice in
replacements are used. Add about one hour of labor if you have to have a
mechanic do it.  The front sensor is more likely to be the culprit than the
rear one. It seems that the most common failure code that basically says
that the cat efficiency is not good can be either the O2 sensor (usually
front) or the cat. Often an additional problem with the ignition coil pack
on 99 and 00s can be part of the problem. The coil pack issue is well
documented in the Miata.net forums, and can be difficult to diagnose.  On
California spec cars, the front precat and O2 sensor are the most common
failure items. One string of logic is that the defective coil pack causes
poor combustion, the contaminants screw-up the pre cat or cat, and the O2
sensor fails prematurely as well.  I've had this happen on my fed spec 99.