I hope you're being honest. If that's the case, you can't convince him of the truth if he won't believe you. Just tell him "I'm not doing that and if you can't trust me I can't stay in this relationship".
As long as you haven't been doing any of these things he's accusing you of, him trusting you is his decision. And if the relation is of any value to him, he'll stop accusing you and give you an opportunity.
Let him go through your phone to check your messages and give him your passwords to Facebook and twitter to show him that you have nothing to hide and that you are being honest. Well that's just my opinion
He's not mature enough to be in a relationship. His insecurities are his problem not yours. If you've been a couple for a few months and he doesn't trust you, time to move on. Why stay in a relationship based on lies and accusations, you can get that anywhere. The key to a solid relationship is honesty & trust, without it you're just wasting your time. He's calling you a cheater & a liar, bye.