The moral reasoning behind restraining the government from denying speech and other rights is that the government is a monopoly (in this case of force). This reasoning extends to restrictions on the ability of other monopolies, such as local utilities, to deny service.
If every service provider with the capacity to serve the needs of Website X were to deny service, the DE FACTO effect is precisely the same as a single monopoly doing so. In which case, the moral reasoning behind restraint of government and other monopolies comes into play.