A measurement of vector boson (V) production in conjunction with a D*(2010)+ meson is presented. Using a data sample corresponding to 9.7 fb-1 of proton-antiproton collisions at center-of-mass energy s=1.96 TeV produced by the Fermilab Tevatron, we reconstruct V+D*+ samples with the CDF II detector. The D*+ is fully reconstructed in the D*(2010)+→D0(→K-π+)π+ decay mode. This technique is sensitive to the associated production of vector boson plus charm or bottom mesons. We measure the ratio of production cross sections σ(W+D*)/σ(W)=[1.75±0.13(stat)±0.09(stat)]% and σ(Z+D*)/σ(Z)=[1.5±0.4(stat)±0.2(stat)]% and perform a differential measurement of dσ(W+D*)/dpT(D*). Event properties are utilized to determine the fraction of V+D*(2010)+ events originating from different production processes. The results are in agreement with the predictions obtained with the pythia program, limiting possible contribution from non-standard-model physics processes.