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You could use either. They are both weaker in intensity than stare which implies a hard and deliberate action.
Thanks, Anglika, for your reply.
For the sake of clarity, would you mind giving a few samples of gaze/peer at so that I can compare their differences? Thanks again.
Thanks, Anglika, for your reply.
For the sake of clarity, would you mind giving a few samples of gaze/peer at so that I can compare their differences? Thanks again.
I was annoyed when he opened the door and peered into my room.
We sat together and gazed into the stars.
Peer > when you look with scrutiny/curiosity
Gaze > fix your eyes with a steady and intent look
gaze/stare at me like that 
Re: gaze/stare at me like that 
Linear Mode
