Ideally, it wouldn't with a level cue. But most shots are performed with an elevated cue because of the rail. As such, the cueball does curve. The phenomena is called "swerve", or "mini-masse", or just plain "curve". The reason it does this is because the spin axis is tilted forward by the same amount that the cue is non-level. This causes the front of the spinning ball to swipe across the cloth.
When the balls are fairly close to each other and a firm stroke is used, the amount if curve on the way to the object ball is negligible to non-existent. It just doesn't have the time to curve to any appreciable extent and is partly airborne so it's not contacting the cloth.
Even with a level cue the cueball will curve ever so slightly because of the cloth's rolling resistance. In effect, the axis is tilted with respect to the cloth even if it's perfectly perpendicular to the slate. But this curve is so small it's hard to measure and doesn't really affect the shot in any practical sense. It can only be seen after some collisions when the ball still has a lot of spin but very little speed.
Swerve has been discussed so much and is so well accepted that you would have to come up with a demonstration or knockout argument to prove it's not true. A lot of people would be surprised.
Jim