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Well you have two choices.
1. Do you have any objection to me watching you.
2. Do you have any objection to my watching you.
Not much difference but 1 does put the focus on 'me' while 2 puts the focus on 'watching.'
- I don't want him watching me while I paint.
- Do you have any objection to me watching you?
- No, you I don't mind, it's just him I can't stand.
As for the grammar of the 'my,' it's claiming possession of the action: my watching, my reading, my writing.
I am reading this book. This betters my soul.
My reading this book betters my soul.
I am watching you paint. Do you have any objection to that?
Do you have any objection to my watching you paint?