The wild-type SARS-CoV-2 has continuously evolved into several variants with increased transmissibility and virulence. The Delta variant which was initially identified in India created a devastating impact throughout the country during the second wave. While the efficacy of the existing vaccines against the latest SARS-CoV-2 variants remains unclear, extensive research is being carried out to develop potential antiviral drugs through approaches like in silico screening and drug-repurposing. This study aimed to conduct the docking-based virtual screening of 50 potential phytochemical compounds against a Spike glycoprotein of the wild-type and the Delta SARS-CoV-2 variant. Subsequently, molecular docking was performed for the five best compounds, such as Lupeol, Betulin, Hypericin, Corilagin, and Geraniin, along with synthetic controls. From the results obtained, it was evident that Lupeol exhibited a remarkable binding affinity towards the wild-type Spike protein (−8.54 kcal/mol), while Betulin showed significant binding interactions with the mutated Spike protein (−8.83 kcal/mol), respectively. The binding energy values of the selected plant compounds were slightly higher than that of the controls. Key hydrogen bonding and hydrophobic interactions of the resulting complexes were visualized, which explained their greater binding affinity against the target proteins—the Delta S protein of SARS-CoV-2, in particular. The lower RMSD, the RMSF values of the complexes and the ligands, Rg, H-bonds, and the binding free energies of the complexes together revealed the stability of the complexes and significant binding affinities of the ligands towards the target proteins. Our study suggests that Lupeol and Betulin could be considered as potential ligands for SARS-CoV-2 spike antagonists. Further experimental validations might provide new insights for the possible antiviral therapeutic interventions of the identified lead compounds and their analogs against COVID-19 infection.