I read the following:
The next day as I watched them get married...
How does it differ from the following? Is one more eloquent and accurate than other?
The next day as I watched them getting married...
It's largely a matter of stylistic choice. There can sometimes be a nuance of meaning involved - for example...
...where the second one implies multiple blows more strongly than the first.
Feasibly a case can be made for saying "I watched him light the fire" attaches more importance to the fact that an act was performed (with the implication that the fire having been lit is important later). In contrast to "I watched him lighting the fire", which might be said to concentrate on the ongoing performance of the act (with the implication that the process itself is "interesting"). Any such distinction is tenuous at best, and would really be better discussed on writers.se
My $0.02 worth,
Implies the marriage ceremony happened in the past and is complete.
Implies the ceremony began on the next day, but might be continuing though the watcher may not still be watching. I believe this is called an imperfect tense.
Technically, you could draw a shade of distinction between the two: "I watched them get married" places more of a focus on the action, while "I watched them getting married" places more of a focus on the people who are performing the action.
In this particular action, it feels like there is really no difference between the two, and in the general case it's very clear from context where the focus was: