Traditional wound dressings often cannot treat wounds caused by bacterial infections or other wound types that are insensitive to these wound treatments. Therefore, a biodegradable, bioactive hydrogel wound dressing could be an effective alternative option. The purpose of this study was to develop a hydrogel membrane comprised of sodium alginate, polyvinyl alcohol, acrylic acid, and gallic acid for treating skin wounds. The newly developed membranes were analyzed using Fourier transform infrared spectroscopy (FTIR), thermogravimetric analysis (TGA), differential scanning calorimetry (DSC) and scanning electron microscopy (SEM), X-ray diffraction analysis (XRD), sol-gel fraction, porosity, mechanical strength, swelling, drug release and data modelling, polymeric network parameters, biodegradation, and antioxidation (DPPH and ABTS) and antimicrobial activity against Gram-positive and negative bacteria. The results revealed that hydrogel membranes were crosslinked successfully and had excellent thermal stability, high drug loading, greater mechanical strength, and exhibited excellent biodegradation. Additionally, the swelling ability and the porosity of the surface facilitated a controlled release of the encapsulated drug (gallic acid), with 70.34% release observed at pH 1.2, 70.10% at pH 5.5 (normal skin pH), and 86.24% at pH 7.4 (wounds pH) in 48 h. The gallic acid-loaded hydrogel membranes showed a greater area of inhibition against Pseudomonas aeruginosa, Staphylococcus aureus, and Escherichia coli bacteria as well as demonstrated excellent antioxidant properties. Based on Franz cell analyses, the permeation flux of the drug from optimized formulations through mice skin was 92 (pH 5.5) and 110 (pH 7.4) μg/cm2·h−1. Moreover, hydrogel membranes retained significant amounts of drug in the skin for 24 h, such as 2371 (pH 5.5) and 3300 µg/cm2 (pH 7.4). Acute dermal irritation tests in rats showed that hydrogel membranes were nonirritating. Hydrogel membranes containing gallic acid could be an effective option for improving wound healing and could result in faster wound healing.