'See' is a sense. We have five senses - sight, hearing, touch, taste and smell. If I can 'see' you it means that you are visible to me. To 'see' is an ability, not an action. To see is involuntary
'I am watching you' and 'I am looking at you' are both conscious actions. They are voluntary. They both mean that I am paying attention to you.
If you say 'I am watching you', the person is probably moving, or I'm thinking that they might move. If you say 'I am looking at you', the person is probably not moving.