I've read The Right Stuff, yes. Good film, too, surprisingly. And yes, it's a valid point of reference in this case, even if it deals with the Mercury 7 instead of the New Nine.
I do believe NASA astronauts were very in control of most of the aspects of their life, but after Apollo 1, they all took safety even more seriously than before (see the return of Gunther Wendt as an example of increased safety) and I believe they would not risk the integrity of a space suit with something as silly as a watch they could not even use on the moon.
Also, seeing as they had to remove clothing to put their suits on, I would be surprised if they didn't take a watch off. David Scott replaced his Omega with a Waltham, wearing it on the outside of his suit with the Omega's velcro strap on EVA 3.
I would concede that a watch other than the Speedies and Scott's Waltham could well have been worn in the LEM, but almost certainly not on an EVA.