Nickel-based superalloy Inconel 625 is considered a hard-to-cut material due to characteristics such as high hardness, strain hardening behavior, and affinity with tool materials. It is challenging to machine through micro-holes of less than Ø 500 µm with the desired quality on Inconel 625 using the mechanical micro-drilling (MMD) process under dry conditions. The machining parameters must be selected optimally to enhance hole quality and to avoid frequent drill breakages. In this study, through micro-holes are machined on Inconel 625 by the MMD process using solid carbide micro-drills under dry conditions using a peck drilling strategy. Experiments are carried out by varying machining parameters such as spindle speed (8000 rpm, 11000 rpm, and 14000 rpm), feed (5 µm/rev, 7.5 µm/rev, and 10 µm/rev), and drill diameter (Ø 300 µm, Ø 400 µm, and Ø 500 µm) at three levels based on full factorial design. Thrust force and hole quality features such as exit burr height, radial overcut, and taper angle are measured as output responses. The mean effect plots are used to study the influence of machining parameters on output responses. The drill diameter had a significant effect on thrust force and radial overcut. Whereas feed and spindle speed had a major influence on exit burr height and taper angle, respectively. Finally, multi-response optimization is carried out using the grey fuzzy logic method, and an optimal machining parameter setting for multiple responses is ascertained.