Detect damage to device after dropping it?
There is no diagnostic software AFAIK (as far as I know ). If there was, you would have heard of it

While things seem to be fine except for the screen , you can never be sure. The insides of phone are densely packed to maintain smaller thickness, size etc and it is quite possible that something else is also affected. 

See sites like iFixit  for help of you wish to do it on your own. You can never be sure that screen is the only damaged part

It's up to you but service centre may be an easier and better option, of course at higher cost and time (and missing out on the fun of doing it yourself :)