As for the original question, "Can an eagle kill a deer?", the answer is yes. Eagles are very powerful and dangerous birds, even when trained for falconry. The gripping strength of their feet is unbelievable. An average human hand has the strength of squeezing a respectable 20 psi. But the grip of the golden eagle is capable of more than 500 psi. (I know, I used to hunt with one.)
Very hungry eagles will often attack, and attempt to kill, quarry much larger than itself. This is a natural survival instinct. Hunger makes an eagle hunt. It will normally hunt small, easy prey. If an eagle is unfed for 2-3 (or more) days (weight loss is determining factor) the bird goes into a condition called "yarak". This is a natural behavior that falconers use to their advantage when hunting with birds of prey. Yarak prompts a raptor to attack and kill with a ferocity five times greater than its "normal" hunting manner. When a wild eagle is pushed into this condition it will most certainly attack, and is capable of killing, deer and even larger prey.