No, many software engineers don't touch the web. Things like programming embedded microprocessors, building operating systems, writing compilers, data mining, AI, integrating into banks, writing software that controls planes / cars, realtime software, software written in ADA for example that deals with very critical things like manned space travel or even nuclear power stations, robotics, 3D vision applications doesn't require web and even mobile and desktop applications doesn't require web if you're building native applications
Web is a nice-to-have skill, although very in demand skill if you don't have any of the other specialities