Hm, odd that they simply dropped the charges.
You're right, though, it's his issue, not yours. He's probably indignant that he was inconvenienced by being arrested and having to go to court!
His dad doesn't live with you, right? Have you talked to him about your plan to make your son leave? Is he supportive of that?
Make sure you get good advice on how to get him out of your home--nothing worse than going through all kinds of hoops only to find out you did it the wrong way. You could call your landlord/tenant court to find out the procedure. They may have a packet of forms you can use. You might also find out whether the procedure is different for evicting an adult family member as opposed to a rent-paying tenant.