According to dict, watch is "look at or observe attentively, typically over a period of time," so no. Merawat is "to take care," "to look after," and "to nurse."
BTW perawat (per+rawat) = nurse. The meaning of "merawat" should be clearer for you with this :)
I like Velonisia's explanation. Colloquially, you could translate it as watch in English; however, translating it back into Indonesian would produce errors. Best try to keep to translations that are reversible. That way you are less likely to use the same word in a different context incorrectly.