Translation:I have asked him if he wanted a watch.
As an additional consideration, the proper tense in this clause should be the imperfect subjunctive (Gli ho chiesto se volesse un orologio), because asking wether/if someone wants something infers uncertainty.
However, being this clause an indirect question, namely an indirect interrogative clause, in colloquial Italian the use of the indicative mood is now considered legitimate. See also this recent discussion about the same type of clause: https://forum.duolingo.com/comment/27503776.
So i understand the past perfect in the first part of the sentence (i asked him) since it is a completed action at a specific time. But what gives the indication that you use the imperfect later in the sentence? Would the past perfect also work to state that "he wanted a watch?" (lui ha voluto)
I think the imperfetto would normally be used because we are talking about an unfinished state (his desire to have a watch) rather than a completed action (he wanted a watch, but not any more?)
As PietroFormano wrote, wanting is a lasting condition, thus unfinished by the time setting of the sentence. Such condition requires the imperfect tense.
A literal translation of the Italian sentence would be "I have asked him if he was wanting a watch", but this would sound odd.
In a similar way, "he was on the train" translates as lui era sul treno (not fu sul treno / è stato sul treno), as being in some place is a lasting condition (not a finished action).
The literal translation is "he was being on the train", but this would sound odd, so the simple past is used.