He's right, you shouldn't check his phone, but then again you shouldn't HAVE to check his phone.
Sounds like, to me, that you should spend way less time on this boy and his games and more time looking for someone who is going to respect YOU and your relationship.
If you've confronted him about it and he hasn't stopped because it makes you uncomfortable, then he's being disrespectful of you and your relationship. This whole situation reminds me of Beyonce's - If I Were A Boy...
Getting respect from him or any relationship should be a given... and here it seems as if it's not. It seems like he's just stringing you along, because he knows you're going to put up with what ever he does. Well, the simple fact is that if you abuse it... you lose it. I.E. If you are going to waste your time abusing someone you are supposedly in a relationship, that you are supposed to care for and love (whether the abuse be mentally, physically, emotionally... what ever) you will lose that person in the end.
I would sit down with him, tell him that it truly bothers you and that you would appreciate it if he stops doing that to you because you really don't deserve it. If he decides that you're being "rediculous" or "jealous" or anything that is here, there ore in between... then tell him that you would rather spend time on finding someone to spend your life and your time with who is going to respect you as a person than to let him keep disrespecting you.
Above all, the worst thing to do is to get really upset and angry. If you don't communicate your feelings to him calmly and in an adult manner, then he will take those high running emotions and use them against you. Making you feel bad about yourself, making the situation way worse than it needs to be... When you're ready to talk to him calmly and rationally, then go ahead and tell him.
Hopefully this helped! Good luck and I hope everything works out for you in one way or another!