In this study, zinc (Zn)- and copper (Cu)-doped 13-93B3 borate mesoporous bioactive glasses (MBGs) were successfully synthesized using nitrate precursors in the presence of Pluronic P123. We benefited from computational approaches for predicting and confirming the experimental findings. The changes in the dynamic surface tension (SFT) of simulated body fluid (SBF) were investigated using the Du Noüy ring method to shed light on the mineralization process of hydroxyapatite (HAp) on the glass surface. The obtained MBGs were in a glassy state before incubation in SBF. The formation of an apatite-like layer on the SBF-incubated borate glasses was investigated by X-ray diffraction (XRD) and scanning electron microscopy (SEM). The incorporation of Zn and Cu into the basic composition of 13-93B3 glass led to changes in the glass transition temperature (Tg) (773 to 556 °C), particle size (373 to 64 nm), zeta potential (−12 to −26 mV), and specific surface area (SBET) (54 to 123 m2/g). Based on the K-means algorithm and chi-square automatic interaction detection (CHAID) tree, we found that the SFT of SBF is an important factor for the prediction and confirmation of the HAp mineralization process on the glasses. Furthermore, we proposed a simple calculation, based on SFT variation, to quantify the bioactivity of MBGs. The doped and dopant-free borate MBGs could enhance the proliferation of mouse fibroblast L929 cells at a concentration of 0.5 mg/mL. These glasses also induced very low hemolysis (<5%), confirming good compatibility with red blood cells. The results of the antibacterial test revealed that all the samples could significantly decrease the viability of Pseudomonas aeruginosa. In summary, we showed that Cu-/Zn-doped borate MBGs can be fabricated using a cost-effective method and also show promise for wound healing/skin tissue engineering applications, as especially supported by the cell test with fibroblasts, good compatibility with blood, and antibacterial properties.