We present a measurement of relative partial widths and decay rate CP asymmetries in K-K+ and pi(-)pi(+) decays of D0 mesons produced in pp collisions at sqrt[s]=1.96 TeV. We use a sample of 2x10(5) D(*+)-->D0pi(+) (and charge conjugate) decays with the D0 decaying to K-pi(+), K-K+, and pi(-)pi(+), corresponding to 123 pb(-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 Gamma(D0-->K-K+)/Gamma(D0-->K-pi(+))=0.0992+/-0.0011+/-0.0012, Gamma(D0-->pi(-)pi(+))/Gamma(D0-->K-pi(+))=0.035 94+/-0.000 54+/-0.000 40, A(CP)(K-K+)=(2.0+/-1.2+/-0.6)%, and A(CP)(pi(-)pi(+))=(1.0+/-1.3+/-0.6)%, where, in all cases, the first uncertainty is statistical and the second is systematic.