No. It can affect the bandwidth, but it takes more than what you'll find on a mobile antenna. Of course if you do slam a radiator with enough raw power, you can cause issues, but unless you are doing something really silly, you'll never run into that.
I agree with what mmckenna said but want to add one minor point. A flimsy antenna will bend quite a bit in the wind and affect the polarity, while a stiffer antenna will remain more vertical (not really much difference in actuallity, but you did ask if it "has anything to do" not "has any real effect on") so a thicker antenna rod may keep the correct polarity a bit better. Will this difference cause a good signal to be unreadable, probably not, and even if it did, you could always slow down and correct the problem.
The other side of this coin is that a flimsy antenna will more easily handle hitting overhanging trees while a stiffer one may bend or break more easily, depending on what was struck and at what speed it was struck. This is one reason most of the old style 102" CB whips had a large spring near the mount.