Detect damage to device after dropping it? I dropped my Pixel. The screen is cracked and doesn't light up.
But the phone still works - it rings and vibrates when I get a call. 

I'm considering replacing the screen myself, but before I buy the part, I'm wondering if there is any way to have confidence that the screen is the only problem.

Can I connect my phone to the computer via USB, and then run some kind of diagnostic software from the computer, to tell me exactly which parts are damaged besides the screen (if any)?
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 :)