Two newly synthesized coumarin–palladium(II) complexes (C1 and C2) were characterized using elemental analysis, spectroscopy (IR and 1H-13C NMR), and DFT methods at the B3LYP-D3BJ/6-311+G(d,p) level of theory. The in vitro and in silico cytotoxicity of coumarin ligands and their corresponding Pd(II) complexes was examined. For in vitro testing, five cell lines were selected, namely human cervical adenocarcinoma (HeLa), the melanoma cell line (FemX), epithelial lung carcinoma (A549), the somatic umbilical vein endothelial cell line (EA.hi926), and pancreatic ductal adenocarcinoma (Panc-1). In order to examine the in silico inhibitory potential and estimate inhibitory constants and binding energies, molecular docking studies were performed. The inhibitory activity of C1 and C2 was investigated towards epidermal growth factor receptor (EGFR), receptor tyrosine kinase (RTK), and B-cell lymphoma 2 (BCL-2). According to the results obtained from the molecular docking simulations, the inhibitory activity of the investigated complexes towards all the investigated proteins is equivalent or superior in comparison with current therapeutical options. Moreover, because of the low binding energies and the high correlation rate with experimentally obtained results, it was shown that, out of the three, the inhibition of RTK is the most probable mechanism of the cytotoxic activity of the investigated compounds.