We present a measurement of relative partial widths and decay rate CP asymmetries in K(sup -)K(sup +) and (pi)(sup -)(pi)(sup +) decays of D(sup 0) mesons produced in p(bar p) collisions at (radical)s = 1.96TeV. We use a sample of 2 x 10(sup 5) D*(sup +) (yields) D(sup 0)(pi)(sup +) (and charge conjugate) decays with the D(sup 0) decaying to K(sup -)(pi)(sup +), K(sup -)K(sup +), and (pi)(sup -)(pi)(sup +), corresponding to 123 pb(sup -1) of data collected by the Collider Detector at Fermilab II experiment at the Fermilab Tevatron collider. No significant direct CP violation is observed. We measure (Lambda)(D(sup 0) (yields) K(sup -)K(sup +))/(Lambda)(D(sup 0) (yields) K(sup -)(pi)(sup +)) = 0.0992 (+-) 0.0011 (+-) 0.0012, (Lambda)(D(sup 0) (yields) (pi)(sup -)(pi)(sup +))/(Lambda)(D(sup 0) (yields) K(sup -)(pi)(sup +)) = 0.03594 (+-) 0.00054 (+-) 0.00040, A(sub CP) (K(sup -)K(sup +)) = (2.0 (+-) 1.2 (+-) 0.6)%, and A(sub CP) ((pi)(sup -)(pi)(sup +)) = (1.0 (+-) 1.3 (+-) 0.6) %, where, in all cases, the first uncertainty is statistical and the second is systematic.