How to stop Google from making site the top search result

Question

I created a website for a clinic that's part of a large hospital. Unfortunately when someone googles the name of the large hospital, the clinic's website with address, map and phone number, comes up first on the search results. So the poor clinic receptionist is fielding a lot of phone calls from people thinking the clinic's number is the hospital's general phone number.
So first I removed the name of the hospital (except in an image) from the home page, and took the hospital name out of the address text. I tried using Google's URL removal tool to tell them to stop using the clinic website for the hospital name search. But each request there is marked denied after a few days. Even tried the Google option to have them reindex the site. So far nothing has worked, and it's been a month.
So does anyone have any other ideas to try and get Google to de-rank or remove the clinic site from the hospital name search results?

Answer 1

Thanks for the robot.txt idea. I didn't know about that.
It seems that won't work though, as I still want all pages to be indexed so people can find the site when they are looking for just the clinic. It's just stopping the query for the hospital from returning the site.
But I've now noticed that the actual site link is further down the search results, but the tiny map and address that show up at the top of the query ( i.e. when you search for a business) is being taken from the clinic address and website.
I'm wondering if it's because I used the Google Map widget in the website. I'm going to try and replace the map widget with a gif of the map and see if that helps. Open to other ideas here too.