No, in water, the spacetime is not significantly warped, certainly not warped enough to slow down light. The different speed of light in a medium compared to vacuum comes from the interaction between the photons and the medium. The photon then, effectively, becomes a hybrid "photon-medium excitation" which is an effective particle with an effective speed lower than the vacuum speed of light.
Maybe not from our point of view, but from the POV of the photon the water is moving with c_0 through the photon, so certainly relativistic effects apply.
No, in water, the spacetime is not significantly warped, certainly not warped enough to slow down light. The different speed of light in a medium compared to vacuum comes from the interaction between the photons and the medium. The photon then, effectively, becomes a hybrid "photon-medium excitation" which is an effective particle with an effective speed lower than the vacuum speed of light.