Do you have to pay a monthly service to use the Apple Watch?

No.

But without an iPhone to pair it with, it is just a fancy timepiece. iPhone does require a cellular plan.

ChitlinsCC wrote:

But without an iPhone to pair it with, it is just a fancy timepiece.

Without an iPhone, it's not even a fancy timepiece. You can't set it up.

I am again humbled...

[the days of having every widget Apple makes is over for me, I fear]

best regards, Ma'am

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now that I think about it, the term I remember being used was not timepiece, but rather "Jewelry" 😉