Is what the email service does by displaying friends' activities on the homepage considered spying if their email is among my contacts?

It is not permissible to view someone else's email without their permission if they dislike it being viewed. This is considered espionage, and espionage is forbidden, as Allah Almighty says: "And do not spy," and as the Prophet, peace and blessings be upon him, said: "Do not go around seeking out information and do not spy." Espionage is the search for faults or the investigation of hidden matters.