Electronic many-body correlation effects in one-dimensional (1D) systems such as carbon nanotubes have been predicted to strongly modify the nature of photoexcited states. Here we directly probe this effect using broadband elastic light scattering from individual suspended carbon nanotubes under electrostatic gating conditions. We observe significant shifts in optical transition energies, as well as line broadening, as the carrier density is increased. The results demonstrate the role of screening of many-body electronic interactions on the different length scales, a feature inherent to quasi-1D systems. Our findings further demonstrate the possibility of electrical tuning of optical transitions and provide a basis for understanding of various optical phenomena in carbon nanotubes and other quasi-1D systems in the presence of charge carrier doping.