Why does a bullet bounce off water?

It's nothing to do with surface tension (art least for large objects).
It's simply the force needed to accelerate the water out of the way to allow the object to sink.

Imagine a bullet bouncing off another bullet, or metal armour. No problem accepting that, it's just Newton's laws and momentum. well water also has mass and needs a force to accelerate it in exactly the same way - the only difference in bouncing a bullet, or a stone, or a bomb, is the speed and angle and how much water you need to move and how fast.

I'm not sure at what speed/pressure the viscosity becomes a factor, has anyone tried skimming stones off super-fluid helium?

The mechanism is explained, e.g., in W. Johnson, Int. J. Impact Engng, Vol.21, Nos 1-2, pp. 15-24 and 25-34. 1998.

The following main assumptions are used to derive the approximate Birkhoff formula for the critical ricochet angle for a spherical projectile:

(i) The pressure $p$ on a spherical surface element along its outward drawn normal is $\rho u^2/2$; u is the forward speed of the sphere resolved along the normal.

(ii) The pressure applies only to those parts of the sphere which are immersed below the undisturbed surface of the water. The effect of the splash on the sphere is considered not to contribute any pressure.

Thus, I believe, surface tension is negligible.