Yoshio, in addition, see is a verb of perception; e.g., see, hear, taste, feel, smell. The action isn't acted out, as is the case with the verb watch; it's perceived, which means to become aware of directly through any of the senses. So,
1a. I watched (the) soccer (game) on TV. <act>
1b. I saw (the) soccer (game) on TV. <perception = act>
See and watch are synonymous when the event involves perception, as in looking at TV (i.e., to see TV) and watching TV. But when the event calls for action, see doesn't fit semantically:
Action: Please
watch my bag (i.e., make sure no one takes it).

Perception: Please
see my bag (i.e., look at it).

Note that, verbs of perception usually don't take -ing, the progressive / continuous tense:
Action: I
was watching (the) soccer (game) on TV.

Perception: I
was seeing (the) soccer (game) on TV.

There are exceptions though. See and other verbs of perception can be used to express an action, as in
Action: Max
is seeing the dentist today (i.e., has an appointment or date with).

All the best