For some reason Polish has two versions of this verb- patrzyć and patrzeć. they have the same present forms but "patrzył" and "patrzał" past forms. While both forms are theoretically correct, Patrzył is much more common past form, while "patrzeć" is much more common
Grammatically, it requires a determiner, which could be either an article or a pronoun, which is why in English both "He was looking at the watch" and "He was looking at his watch" are grammatically correct, though in most contexts (i.e. if he was looking at the watch on his wrist to check the time) the pronoun would sound more natural.
PL swój / jego : ENG his Where in the polish sentence in than "jego" or "swój"??? The correct translation of "On patrzył na zegarek" is "He was looking at the watch". Anyway if it will be "On patrzył na swój zegarek" the correct translation would be "He was looking at his watch". (I am native polish.)
Adding to this point, English uses possessives a lot because singular countable nouns need determiners. In a situation where someone was looking at his watch to check the time (i.e. the most likely or most common situation for "on patrzył na zegarek"), it would not sound at all natural to say "he was looking at the watch" or "he was looking at a watch" (though both are grammatically correct). Since a determiner is required, this leaves "his" as the most natural choice and "he was looking at his watch" is definitely the most natural sentence in this situation. Situations where the other two sentences fit are possible, but much less common.