The spin-gap evolution upon Ca doping in Sr14-xCaxCu24O41 was systematically investigated using inelastic neutron scattering. We discover that the singlet-triplet spin-gap excitation survives in this series with x up to 13, indicating the singlet dimer ground state in these compounds. This observation corrects the previous speculation that the spin gap collapses at x similar to 13 by the NMR technique. The strong intensity modulation along Q(H) in x = 0 gradually evolves into a Q-independent feature in x > 11. This could be attributed to the localized Cu moment magnetism developing into an itinerant magnetism with increasing x. It is a surprise that the spin gap persists in the normal state of this spin-ladder system with metallic behavior, which evidences the possibility of magnetically mediated carrier pairing mechanism in a two-leg spin-ladder lattice.