We calculate the long-range perturbation to the electronic charge density of carbon nanotubes (CNTs) as a result of the physisorption of a water molecule. We find that the dominant effect is a charge redistribution in the CNT due to polarisation caused by the dipole moment of the water molecule. The charge redistribution is found to occur over a length-scale greater than 30 Å, highlighting the need for large-scale simulations. By comparing our fully first-principles calculations to ones in which the perturbation due to a water molecule is treated using a classical electrostatic model, we estimate that the charge transfer between CNT and water is negligible (no more than 10(-4) e per water molecule). We therefore conclude that water does not significantly dope CNTs, a conclusion that is consistent with the poor alignment of the relevant energy levels of the water molecule and CNT. Previous calculations that suggest water n-dopes CNTs are likely due to the misinterpretation of Mulliken charge partitioning in small supercells.