This work aims to improve the properties of poly(lactic acid) (PLA) for future biomedical applications by investigating the effect of montmorillonite (MMT) nanoclay on physicochemical and mechanical behavior. PLA nanocomposite filaments were fabricated using different amounts of MMT (1.0, 2.0, and 4.0 wt.%) and 2 wt.% Joncryl chain extenders. The 3D-printed specimens were manufactured using Fused Filament Fabrication (FFF). The composites were characterized by Gel Permeation Chromatography (GPC), Melt Flow Index (MFI), X-ray Diffraction (XRD), and Fourier-transform infrared spectroscopy (FTIR). The thermal properties were studied by means of Differential Scanning Calorimetry (DSC) and Thermogravimetric Analysis (TGA). Moreover, the hydrophilicity of the PLA/MMT nanocomposites was investigated by measuring the water contact angle. The mechanical behavior of the PLA/MMT nanocomposites was examined with nanoindentation, compression tests, and Dynamic Mechanical Analysis (DMA). The presence of Joncryl, as well as the pretreatment of MMT before filament fabrication, improved the MMT distribution in the nanocomposites. Furthermore, MMT enhanced the printability of PLA and improved the hydrophilicity of its surface. In addition, the results of nanoindentation testing coupled with Finite Element Analysis showed that as the MMT weight fraction increased, as well as an increased Young’s modulus. According to the results of the mechanical analysis, the best mechanical behavior was achieved for PLA nanocomposite with 4 wt.% MMT.