Old content in new url: can i now delete completely the old blog?

Hi!
So, I tried to find the answer in older posts but I am just too terrified to try without being sure.
I changed the URL of my blog keeping exactly the same content of the old one and without discarding the old address (I put the old blog under “private” in the settings).
Now, I have two sites under “My Sites” and only the old one is appearing (blank and with “no results found”) when I google the title (which is the same of the old one).
The question in simple: can I delete completely the old site? Or will this do something to my new now, as they share the same content?

This question is maybe very silly, but thank you so much for the help!
Greta