This paper presents a study on the prospects of functionalizing nanodiamonds (NDs) with aminoacetic acid to obtain high-strength composites based on an epoxy matrix. The impact of the functionalization of the ND surface with aminoacetic acid in various concentrations on the properties of the epoxy composite was assessed. The success of grafting amine onto the ND surface was confirmed by X-ray phase analysis and IR spectroscopy. The results show a significant decrease in the average size of ND particles, from 400 nm for the pristine ones to 35 nm, and the contact angle, from 27° to 22°, with an increase in the specific surface area after treatment with a 5% solution of aminoacetic acid. Reducing the average size of NDs allows them to be better distributed throughout the epoxy matrix, which, as a result of the formation of chemical interaction at the matrix–nanofiller phase interface, can significantly increase the strength of the obtained composite. The addition of NDs treated with aminoacetic acid ensures an increase in the deformation-strength properties of epoxy composites by 19–23% relative to an epoxy composite containing the pristine NDs. Moreover, the presence of functionalized NDs significantly influences the structure and thermal stability of the epoxy nanocomposite.