Needless to say, this answer contains spoilers. Therefore, I have chosen not to spoiler it.
No, Okabe could not have saved Mayuri, as he found out by himself the hard way.
Mayuri was meant to die in the alpha world line; her death is what is called a "convergence point". No matter how Okabe manipulates the alpha world line, creating a new timeline in the process, every timeline belonging to the alpha world line would eventually "converge" to make that one event happen without fail: Mayuri's death.
And Okabe did try to save Mayuri countless times. If breaking her watch were enough to save her, then the convergence point would have been the event where her watch breaks. Thus, in at least one timeline, Mayuri's watch would've broken without Mayuri dying.
Edit (after playing the VN):
Pockety (the watch) actually never broke in the VN, or if it did, it was considered so minor it was never mentioned even once.
It is likely that the scene where it breaks was added for the anime to stylize Mayuri's death, in order to imply her death without actually depicting it. Remember that Mayuri died in several — at times gruesome — ways, such as getting shot in the head, or getting crushed by a subway train. The anime team likely didn't want to make the show too gory, as it would not have fit with the rest of the series very well. The theory isn't too far-fetched, since watches symbolize time, and time symbolizes life. Since Mayuri carries Pockety (a memento of her late grandmother's) at all times, it may as well signify her own life; its breaking meaning her death.
Additionally, despite carrying its own name (Pockety), the watch was never important in the VN in the first place. I figure the anime team wanted to give it meaning in some form, which I'll admit is a nice touch on their part.
In any case, the watch breaking was never meant to be important at all.
As for Makise Kurisu's case. Okabe could save her by making her lie in a pool of blood, because that was the only thing he ever established. When he first saw her, he never actually bothered to check if she was dead in the first place, so convergence might not have been her death, but her lying in a pool of blood instead.
This was of course a gamble. There was no guarantee it would've worked, since they didn't know which event was the actual convergence point: Makise's death, or her lying in a pool of blood (or both). The fact that she survived was merely a lucky shot that could've gone wrong.
In retrospect, Christina could've died of a heart attack after getting tased by Okabe. Mayuri died of heart attacks quite often too, when nothing else was killing her. Although Mayuri's death was never specified in those instances where she simply "collapsed, breathing heavily, clutching her chest", and turned out to be dead, I'll just assume it was a heart attack.
Okabe simply got damn lucky with his assistant.