Which two are correct? ;) The first and last are both OK. The last one could suggest that he watches for longer than the first one.

Numbers 1 and 4. There are two patterns we use after sensing verbs like "watch" "notice" "see" "smell" "hear" "feel" and "taste". A: watch [sentence with "bare infinitive"] | (a "bare infinitive" is an infinitive without "to": to swim (infinitive)-->swim (bare infinitive)) I saw him run. I heard him yell. I noticed him lie. B. A slightly less common pattern is [sensing verb] [noun] [verbing] I saw him swimming. Here, the main sentence is "I saw him." Then, I add a participial verb to modify "him". If you "see someone verbing", it will stress that that the verb is a process that takes a long time. "I saw him writing a book" would usually be a better choice than "I saw him write a book", because you just saw him in the middle of the activity--you didn't observe the entire activity. "I saw him sneeze" would mean you saw him sneeze one time. "I saw him sneezing" would mean you saw him sneeze several times, and concluded that he was sick. If you say "I watched my son play", you were there when he started to play, and you were there when he finished playing. If you say "I watched my son playing", you noticed that he was playing, but you didn't wee him start to play or stop playing.

4.He watches his son playing. (for making sentence need only one principle verb.watches used as main verb and playing used as gerund.