We present a measurement of relative partial widths and decay rate CP asymmetries in K−K+ and π−π+ decays of D0 mesons produced in pp collisions at s√=1.96  TeV. We use a sample of 2×105 D*+→D0π+ (and charge conjugate) decays with the D0 decaying to K−π+, K−K+, and π−π+, 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 Γ(D0→K−K+)/Γ(D0→K−π+)=0.0992±0.0011±0.0012, Γ(D0→π−π+)/Γ(D0→K−π+)=0.035 94±0.000 54±0.000 40, ACP(K−K+)=(2.0±1.2±0.6)%, and ACP(π−π+)=(1.0±1.3±0.6)%, where, in all cases, the first uncertainty is statistical and the second is systematic.