I didn't approve my colleague's annual leave request.
Why didn't you approve your colleague's annual leave request?
We're short handed and can't have any employee out on leave right now.
I'm sure it was a difficult decisions but as a supervisor you have the right to deny or approve annual leave requests. Explain to your employee that you're short-staffed and can't approve their leave request. Ask them to pick an alternative date for their leave.
I don't care if my employee likes it or not. They should always listen to me.
You shouldn't look at employees as your slave. If you mistreat them like this, they will end up quitting and work somewhere else.