I hate to ask this, but is he cheating?
From personal experience, the ones the most jealous are the ones doing the act they are accusing you of. And I sadly see the "playful" way he asks you if the baby is his as a HUGE warning sign...

He went through your messages? This isn't playful anymore. Something is up, and he's making you feel bad about his insecurity. Do you have friends or family that could possibly help you?

I'm not trying to be negative, but I've witnessed this behavior personally and other friends going through it. Usually the arguments escalate, because the one who is being accused gets sick of it and the other person gets angry because of guilt. I hope, in this case, that I'm wrong.

Just - be careful. And if you have friends and family, I would keep in touch as much as possible.