The phase velocity of light changes significantly when it passes from a vacuum into air or from air into water. That change in velocity is what leads to refraction.

Significant in what way? In terms of Relativity, no. Relativity is based on c being invariant, which is the speed light travels in a vacuum. So for example, in a vacuum, two observer moving relative to each other will both measure the same beam of light as moving at c relative to themselves. In water however, this would not be the case. Someone as rest with respect to the water would measure the speed of a beam light moving at one speed relative to himself and someone moving at v relative to the water would measure a different value. But even in this case, c is instrumental. For the person at rest with respect to the water measures a speed of c/n, where n is the index of refraction. The other observer will measure a speed of (v+c/n)/(1+v/(nc))* *Velocity addition formula is w = (v+u)/(1+ vu/c^2). In this case, u = c/n, which causes the equation to reduce to the above form.

Does it ever occur to you to google something before you post your question here? Are you asking people's personal opinions? Define "significant" in this context, please.