The verb "beobachten" contains the word "Obacht" which means "attention" in English. This is quite close to the meaning of the English verb "to watch", as a watchman has to pay attention, but it has nothing to do with clocks or watches in German.
We use the word "Uhr" for any device that shows the time, no matter whether it is a big clock, a watch on somebody's wrist or a digital Display. Only if you want to make a difference, you specify that it is a "Wanduhr", "Turmuhr", "Armbanduhr", "Digitaluhr" or whatever.
When you want to say what time it is, you always say: "Es ist xx Uhr". The word "Stunden" is only used when talking about a duration or the amount of time elapsed after a given point:
"Die Fahrt dauert 6 Stunden"
"2 Stunden nach der Abfahrt macht er eine Pause"
"Jetzt ist er 4 Stunden unterwegs"
"Um 10 Uhr kommt er an".