Well, you did say "require" a watch. I've been hiking nigh on 35 years now, and have never had a "need" for a watch. I will say a watch is desirable, but on most of my hikes - whether in groups or solo - I tend not to carry a watch, except for recently when there was a requirement.
In that case, we had a young hiker who required timely medications.
I also wonder if you mean to distinguish between a real wrist-worn watch and a cellphone. (and what of the new Apple watches...)
Also, "required" is different for some, and not for others. Recording your adventures by logging, for example. Some may require a log to be timestamped, while others just use relative time and sequence of events - time be damned.
For navigation, I suppose if I lost my compass, a watch would come in handy; but I'd have to say that my orienteering skills are not very strong without knowing how to orient a map properly with a watch. So even with a watch, I don't think it would help in this case. Anyway, a compass is a better navigational device than a watch, and a smartphone is better than a compass. As a backup measure, I'd have both a smartphone and a compass. A watch might make a good backup device, but this doesn't seem to fit the definition of "required".
Some here suggest a watch is good for knowing when to take breaks and when to eat, and so on. Here, I think "want" is the more important factor - not "need". Knowing that darkness is going to set in in, say, 2 hours: what does that do for you? Wouldn't you hike until it got dark? Or stop now and enjoy the company or vista for the night?
One need I can think of is in first aid: you apply a tourniquet, you should record the time. (Ah, but this is a recommendation, not an absolute requirement, so not sure if this meets your requirement or not).
On the other hand, if your party wants to split up and meet again at some point in the future, then, having a watch is mandatory: not being on time could be grounds for calling in rescue.
Some say a watch is required for cooking. I don't agree. Who "cooks" on hikes? If your menu includes poached or hard boiled eggs, then perhaps a watch is needed. But all the hikes I've been on, boiling water was needed, and did not require precise timing. So this is a subjective answer: some insist to cook, while others stop at the boiling point.
And then there are some who would like to record their blood pressure and pulse. For medical purposes, I'd say - yes, absolutely. But for recreational purposes - watching your exercise - then no, this fits into the "want" category, not "need".
And finally, there are those who like to keep logs of their adventures. Here, a timepiece would be nice to have, but altogether not "required", I think.
So there you have it: timely medications, the uncommon first aid procedure, medical procedures, and rendezvous. All else seems to me that a watch is for "want" and not "need". Even in navigation.