Nowadays, novel and advanced methods are being sought to efficiently remove dyes from wastewaters. These compounds, which mainly originate from the textile industry, may adversely affect the aquatic environment as well as living organisms. Thus, in presented study, the synthesized ZrO2–SiO2 and Cu2+-doped ZrO2–SiO2 oxide materials were used for the first time as supports for laccase immobilization, which was carried out for 1 h, at pH 5 and 25 °C. The materials were thoroughly characterized before and after laccase immobilization with respect to electrokinetic stability, parameters of the porous structure, morphology and type of surface functional groups. Additionally, the immobilization yields were defined, which reached 86% and 94% for ZrO2–SiO2–laccase and ZrO2–SiO2/Cu2+–laccase, respectively. Furthermore, the obtained biocatalytic systems were used for enzymatic decolorization of the Remazol Brilliant Blue R (RBBR) dye from model aqueous solutions, under various reaction conditions (time, temperature, pH). The best conditions of the decolorization process (24 h, 30 °C and pH = 4) allowed to achieve the highest decolorization efficiencies of 98% and 90% for ZrO2–SiO2–laccase and ZrO2–SiO2/Cu2+–laccase, respectively. Finally, it was established that the mortality of Artemia salina in solutions after enzymatic decolorization was lower by approx. 20% and 30% for ZrO2–SiO2–laccase and ZrO2–SiO2/Cu2+–laccase, respectively, as compared to the solution before enzymatic treatment, which indicated lower toxicity of the solution. Thus, it should be clearly stated that doping of the oxide support with copper ions positively affects enzyme stability, activity and, in consequence, the removal efficiency of the RBBR dye.