Grape stalks (GSs) from winemaking were submitted to a green process to valorise its lignocellulosic biomass that applied subcritical water extraction (SWE) at 170 °C and 180 °C to obtain active extracts and cellulose-enriched fractions. The sum of the total phenolic content of the soluble extract and the solid residue fractions from the SWE exceeded that of the GS, which suggests the generation of compounds with antioxidant properties through SWE. All SWE fractions showed high antioxidant power. The increased temperature promoted the extraction of polyphenolic compounds, enhancing the antioxidant power of both extracts and solid residues. These solid residue fractions were bleached with alkaline hydrogen peroxide solutions (4 and 8% v/v) to purify cellulose. After two bleaching cycles, no notable delignification progress was observed, as the bleaching yield or whiteness index did not significantly change in the further cycles. The first bleaching cycle led to a significant reduction in the lignin content at both SWE temperatures. The cellulose purity was higher in the samples obtained at 170 °C and bleached with 4% alkaline hydrogen peroxide. SWE at 180 °C led to greater cellulose oxidation during the bleaching step regardless of the hydrogen peroxide concentration.