This paper describes the structure-based design and synthesis of two novel square-planar trans-N₂O₂ Cu(II) complexes [Cu(L¹)₂] (1) and [Cu(L²)₂] (2) of 2-((Z)-(4-methoxyphenylimino)methyl)-4,6-dichlorophenol (L¹H) and 2-((Z)-(2,4-dibromophenylimino)methyl)-4-bromophenol (L²H) as potential inhibitors against the main protease of the SARS-CoV-2 and HIV viruses. Copper complexes (1) and (2) crystallize in a monoclinic crystal system with P2₁/n and P2₁/c space groups, respectively [a = 12.4630(12) Å, b = 9.2765(10) Å, c = 12.6425(13) Å, α = 90°, β = 111.863°, γ = 90° and Z = 2 for [Cu(L¹)₂] (1); a = 10.1185(13) Å, b = 10.9809(12) Å, c = 12.5803(13) Å, α = 90°, β = 103.795°, γ = 90°, and Z = 2 for [Cu(L²)₂] (2)]. Single crystal X-ray diffraction studies revealed that both complexes exhibit square-planar geometry (τ = 0). Quantum computational calculations were used for the structure–property relationship. Detailed structural and non-covalent supramolecular interactions in the complexes were investigated by single crystal structure analysis and computational approaches. Hirshfeld surface and 2D fingerprint plots were explored in the crystal structure of the complexes. The strength of the interaction and 3D topology of the crystal packing are visualized through an energy framework. Further, inspired by recent developments to find a structure-based drug design for inhibitors of the SARS-CoV-2 main protease, molecular docking of the copper complexes with the SARS-CoV-2 main protease for COVID-19 was performed. The X-ray crystallographic structures of the main protease of the SARS-CoV-2 virus (PDB ID: 6XBG) and HIV virus (PDB ID: 1JLE and 1UUD) were retrieved from the protein data bank and used as receptor proteins. The molecular docking calculations of complexes (1) and (2) with SARS-CoV-2 virus revealed binding affinities of −9.8 kcal mol⁻¹ and −9.4 kcal mol⁻¹ with inhibition constants of 2.912 μM and 2.813 μM, respectively, at the inhibition binding site of the receptor protein. Besides this, molecular docking against HIV-1 reverse transcriptase (PDB ID: 1JLE) and HIV-1 TAR RNA (PDB ID: 1UUD) were also studied. The molecular docking results also showed that copper complexes with HIV-1 and HIV-1 RNA exhibited good binding affinities and inhibition constants at the binding site of the receptor protein. It was observed that the binding affinities of the copper complexes towards SARS-CoV-2 were comparatively higher than towards the HIV virus. Overall, an in silico molecular docking study suggests the potential role of copper complexes as antivirus drug candidates targeting the SARS-CoV-2 M^pro and HIV protease inhibitors.