The growing demand for cellulosic pulp presents an opportunity to explore alternatives to this material, focusing on utilizing agro-industrial residues. Mango’s tegument is a rich source of cellulose, making it a valuable raw material for manufacturing single-use articles or blends with biopolymers. In this sense, employing conventional alkaline and acid chemical treatments, the mango’s tegument was treated to obtain cellulosic pulp. The teguments were subjected to treatment with alkaline solutions (2% and 4% NaOH w/v) at 80 °C for 1 or 2 h or with an acetic acid solution (1:1 or 1:2 CH3COOH:H2O2) at 60–70 °C for 1 or 2 h. After treatment, an evaluation was conducted to assess the yield, color, chemical analysis, and structural, thermal, and morphological properties. The alkali treatments produced cellulosic pulps with a light color with 37–42% yield and reduced hemicellulose content. The acid treatments produced orange–brown cellulosic pulp with 47–48% yield and higher hemicellulose content. The acid pulps were thermally more stable (maximum decomposition at 348–357 °C) than the alkali pulps (maximum decomposition at 316–321 °C). The crystallinity index demonstrated that both treatments increased the crystallinity of the cellulose pulps compared with the untreated tegument. The thermal stability of cellulosic pulp at the processing temperatures of disposable tableware (50–120 °C) revealed that plates, bowls, trays, and cups could be produced. Another potential application is as a component of blends with biopolymers to make straws or rigid food packaging (trays) with reinforced structures.