Open heavy-flavor hadrons provide unique probes of the medium produced in ultrarelativistic heavy-ion collisions. Due to their increased mass relative to light-flavor hadrons, long lifetime, and early production in hard-scattering interactions, they provide access to the full evolution of the partonic medium formed in heavy-ion collisions. This paper reports two-dimensional (2D) angular correlations between neutral D mesons and unidentified charged particles produced in minimum-bias Au+Au collisions at sNN=200GeV. D0 and D¯0 mesons are reconstructed via their weak decay to Kπ± using the Heavy Flavor Tracker in the Solenoidal Tracker at RHIC experiment. Correlations on relative pseudorapidity and azimuth (Δη,Δφ) are presented for peripheral, midcentral, and central collisions with D0 transverse momentum from 2-10GeV/c. Attention is focused on the 2D peaked correlation structure near the triggered D0 meson, the near-side (NS) peak, which serves as a proxy for a charm-quark-containing jet. The correlated NS yield of charged particles per D0 meson and the 2D widths of the NS peak increase significantly from peripheral to central collisions. These results are compared with similar correlations using unidentified charged particles, consisting primarily of light-flavor hadrons, at similar trigger particle momenta. Similar per-trigger yields and widths of the NS correlation peak are observed. The present results provide additional evidence that D0 mesons undergo significant interactions with the medium formed in heavy-ion collision and show, for the first time, significant centrality evolution of the NS 2D peak in the correlations of particles associated with a heavy-flavor hadron produced in these collisions.