Mineralized poly(ε-caprolactone)/gelatin core-shell nanofibers were prepared via co-axial electrospinning and subsequent incubation in biomimetic simulated body fluid containing ten times the calcium and phosphate ion concentrations found in human blood plasma. The deposition of calcium phosphate on the nanofiber surfaces was investigated through scanning electronic microscopy and X-ray diffraction. Energy dispersive spectroscopy results indicated that calcium-deficient hydroxyapatite had grown on the fibers. Fourier transform infrared spectroscopy analysis suggested the presence of hydroxyl-carbonate-apatite. The results of a viability assay (MTT) and alkaline phosphatase activity analysis suggested that these mineralized matrices promote osteogenic differentiation of human adipose-derived stem cells (hASCs) when cultured in an osteogenic medium and have the potential to be used as a scaffold in bone tissue engineering. hASCs cultured in the presence of nanofibers in endothelial differentiation medium showed lower rates of proliferation than cells cultured without the nanofibers. However, endothelial cell markers were detected in cells cultured in the presence of nanofibers in endothelial differentiation medium.