Zivzik pomegranate (Punica granatum) has recently sparked considerable interest due to its nutritional and antioxidant properties. To evaluate the antioxidant capacities of P. granatum juice, ethanol (EEZP), and water (WEZP) extracts from peel and seed, the antioxidant methods of 2,2′-azino-bis-3-ethylbenzthiazoline-6-sulphonic acid radical (ABTS•+) scavenging, 1,1-diphenyl-2-picrylhydrazyl free radical (DPPH•) scavenging, Fe3+-2,4,6-tris(2-pyridyl)-S-triazine (TPTZ) reducing, Fe3+ reducing, and Cu2+ reducing methods were used. The antioxidant capacities of samples were compared with the most commonly used synthetic antioxidants, i.e., BHA, BHT, α-tocopherol, and Trolox. In terms of setting an example, the IC50 values of EEZP for ABTS•+ and DPPH• scavenging activities were found to be lower than standards, at 5.9 and 16.1 μg/mL, respectively. The phenolic and flavonoid contents in EEZP peel were 59.7 mg GAE/g and 88.0 mg QE/g, respectively. Inhibition of α-glycosidase, α-amylase, acetylcholinesterase, and human carbonic anhydrase II (hCA II) enzymes was also investigated. EEZP demonstrated IC50 values of 7.3 μg/mL against α-glycosidase, 317.7 μg/mL against α-amylase, 19.7 μg/mL against acetylcholinesterase (AChE), and 106.3 μg/mL against CA II enzymes. A total of 53 phenolic compounds were scanned, and 30 compounds were determined using LC-MS/MS. E. coli and S. aureus bacteria were resistant to all four antibiotics used as standards in hospitals.