I feel that the explanation for this needs a picture:

On the left is the situation you have outside of the water. When you look at the watch, the light you see was refracted through the glass, so that the internal angle is less than the critical angle (where total internal reflection occurs). On the right you see the situation where you are observing from inside the water. Key difference is that there is a much smaller refractive index mismatch at the first interface (water/glass) so the light travels almost straight; consequently, it is capable of reaching the second interface at an angle greater than the critical angle, and undergoing total internal reflection.
If we assume the refractive index of the three media (top to bottom) to be $n_1$, $n_2$ and $n_3$, then the condition for total internal reflection can be calculated by working backwards:
$$\sin\theta_{2} =\frac{n_3}{n_2}$$
for the critical angle at the interface between 2 and 3 (where $\sin\theta_3=1$)
$$\frac{\sin\theta_{1}}{\sin\theta_{2}}=\frac{n_2}{n_1}$$
Substituting, we find the angle $\theta_1$ at which the watch will look like a mirror is given by
$$\sin\theta_1 = \frac{n_3}{n_1}$$
In other words - when $n_1$ (the refractive index of the medium with the observer) is greater than $n_3$ (the refractive index of the medium inside the watch) then it is possible that you cannot see the dial.
This problem has been solved by at least one manufacturer of extreme diving watches, by filling the mechanism with a clear fluid with the same refractive index as the (sapphire) "glass":

source: SINN catalog. The watch in the middle of the image has this technology - the others don't. This has the added advantage of making the watch extremely pressure resistant, since the liquid inside is highly incompressible. The watch is rated for 200 bar (2000 m depth). That's not a depth any human should try to dive to.