(abridged) We identify a numerically significant population of heavily obscured AGNs at z~0.5-1 in the Chandra Deep Field-South (CDF-S) and Extended Chandra Deep Field-South by selecting 242 X-ray undetected objects with infrared-based star formation rates (SFRs) substantially higher (a factor of 3.2 or more) than their SFRs determined from the UV after correcting for dust extinction. An X-ray stacking analysis of 23 candidates in the central CDF-S region using the 4 Ms Chandra data reveals a hard X-ray signal with an effective power-law photon index of Gamma=0.6_{-0.4}^{+0.3}, indicating a significant contribution from obscured AGNs. Based on Monte Carlo simulations, we conclude that 74+-25% of the selected galaxies host obscured AGNs, within which ~95% are heavily obscured and ~80% are Compton-thick (CT; NH>1.5x10^{24} cm^{-2}). The heavily obscured objects in our sample are of moderate intrinsic X-ray luminosity [ ~ (0.9-4)x10^{42} erg/s in the 2-10 keV band]. The space density of the CT AGNs is (1.6+-0.5)x10^{-4} Mpc^{-3}. The z~0.5-1 CT objects studied here are expected to contribute ~1% of the total XRB flux in the 10-30 keV band, and they account for ~5-15% of the emission in this energy band expected from all CT AGNs according to population-synthesis models. In the 6--8 keV band, the stacked signal of the 23 heavily obscured candidates accounts for <5% of the unresolved XRB flux, while the unresolved ~25% of the XRB in this band can probably be explained by a stacking analysis of the X-ray undetected optical galaxies in the CDF-S (a 2.5 sigma stacked signal). We discuss prospects to identify such heavily obscured objects using future hard X-ray observatories.
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