The phrase "he watched him" is correct and usable in written English.
You can use it to indicate that someone was watching another person. Example: He watched him carefully as he crossed the street.
He watched him being killed.
He watched him win his debut by five lengths.
He didn't just interview Jobs; he watched him die.
And he watched him fight brain cancer after the disease was diagnosed in 1990.
In the video, a plainclothes officer tells Garner that he watched him sell cigarettes.
Although Marsh never played against him, he watched him often enough to be an admirer.
And with Tadzio's implicit consent, he watches him on the beach and in the hotel, even following him through the streets of Venice.
He also watched him return in 1921, embittered by the violence and chaos that the revolution had unleashed.
And he has watched him wait 10 long years between N.F.L. starts.
He'd watched him dipping under 60 seconds for the 100m and had seen him go sub-2 10 sub-2 10200m.
