The short answer is that no, this is not a viable solution, at least not in the theoretical, anything-can-happen type of world.
The clock of a computer might be accurate to a billionth of a billionth of a second, but the problem isn't the accuracy, it's whether the clocks of the two computers are synchronized.
Here's an experiment. Look at your watch, then ask a random stranger around you what the time is. If your file was written to on your computer when you looked at the watch, and written to on the computer belonging to the person you're asking 1 second ago, would your comparison determine that your file or his/her file was newer?
Now, your solution might work, assuming that:
- You're not going to have to compare milli- or nano-second different clock values
- The clocks of the computers in question are synchronized against some common source
** or at the very least set to be as close to each other as your inaccuracy-criteria allows for
Depending on your requirements, I would seriously look at trying to find a different way of ensuring that you get the right value.