Recent experiments have found that monolayer 1H-TaS2 grown on Au(111) lacks the charge density wave (CDW) instability exhibited by bulk 2H-TaS2. Additionally, angle-resolved photoemission spectroscopy measurements suggest that the monolayer becomes strongly electron doped by the substrate. While density functional theory (DFT) calculations have shown that electron doping can suppress the CDW instability in monolayer 1H-TaS2, it has been suggested that the actual charge transfer from the substrate may be much smaller than the apparent doping deduced from photoemission data. We present DFT calculations of monolayer 1H-TaS2 on Au(111) to explore substrate effects beyond doping. We find that the CDW instability is suppressed primarily by strong S-Au interactions rather than by doping. The S-Au interaction results in a structural distortion of the TaS2 monolayer characterized by both lateral and out-of-plane atomic displacements and a 7 x 7 periodicity dictated by the commensurate interface with Au. Simulated STM images of this 7 x 7 distorted structure are consistent with experimental STM images. In contrast, we find a robust 3 x 3 CDW phase in monolayer 1H-TaS2 on a graphene substrate with which there is minimal interaction.