I think guys just display emotions differently and also do seem to disconnect better in a breakup - it doesn't mean that he doesn't care, but ultimately (mutual or not) he's moving on with his life. As harsh as it sounds, you broke up for a reason and whilst he was a big part of your life, he's not anymore and you need to focus your energy on to moving on as he is doing. There's a chance he may have seen the breakup coming and checked out/prepared himself for it, whilst it may have been more of a shock to you. Clearly there is an emotional attachment there, and you are only going to get rid of that feeling by removing yourself and stop checking in on his life, as harsh as it sounds. Coming from experience, in the early stages/months, you will just sink further down if your focus is still on him and checking up on him, or waiting for the next communication.
And unfortunately, only 2 months in when you've had a sexual relationship and it was more than friends, you are never going to manage just a friendship. We all make the same mistake and are naive with our first breakup, but the biggest healer is time and space. You need to separate your lives and go no contact. Most people take months or years of separation to get to a friendship point - you can't just switch from a relationship back to a friendship, unless there was initially a non-functional relationship beforehand.