"The Watch" is a contraction of 'The Night Watchmen", who were civil employees contracted to patrol the streets at night, calling out the hour, keeping a lookout for fires, checking that doors were locked and ensuring that drunks and other vagrants were delivered to the watch constable. This goes back to biblical times and is pretty much worldwide. Its function has now been mostly replaced by professional policemen.
In England official service of this nature dates back to the early 1200s, in Colonial America Boston established a force of Watchmen in the early 1600s.
Pocket, and later, wristwatches now perform the function of 'telling the hour' that the watchman used to undertake, eg "Four o' the clock and all's well", or 'Six o'clock and a frosty morning!"
That's why a timepiece is called a 'watch'.
