We report on a measurement of CP-violating asymmetries (ACP) in the Cabibbo-suppressed D0→π+π− and D0→K+K− decays reconstructed in a data sample corresponding to 5.9  fb−1 of integrated luminosity collected by the upgraded Collider Detector at Fermilab. We use the strong decay D*+→D0π+ to identify the flavor of the charmed meson at production and exploit CP-conserving strong cc pair production in pp collisions. High-statistics samples of Cabibbo-favored D0→K−π+ decays with and without a D*± tag are used to correct for instrumental effects and significantly reduce systematic uncertainties. We measure ACP(D0→π+π−)=(+0.22±0.24(stat)±0.11(syst))% and ACP(D0→K+K−)=(−0.24±0.22(stat)±0.09(syst))%, in agreement with CP conservation. These are the most precise determinations from a single experiment to date. Under the assumption of negligible direct CP violation in D0→π+π− and D0→K+K− decays, the results provide an upper limit to the CP-violating asymmetry in D0 mixing, |AindCP(D0)|<0.13% at the 90% confidence level.