Does placing the mouse cursor over an image of the Qur'an on a computer, with the intention of doing something, fall under the ruling of an oath, and what is the ruling on swearing to oneself?

Placing one's hand on the Mus'haf or placing the mouse cursor on its image on the computer is not an oath, nor is it considered a sworn affirmation. An oath is only binding when spoken. Swearing by the Qur'an is a binding oath according to the majority of scholars.