Hi there SRKmama, welcome to the forum, I hope you will enjoy it
As to your 'predicament' - I think you are right. if you're his girlfriend you should be introduced. Full stop. No reason he is giving is qualified to undermine your right of being introduced to the lady in question. Nothing has to be 'going on' - but not being introduced is just insulting you. So, you tell him to either quit playing games about this subject (mind you NOT talking with other women, but you're not being introduced formally) or he's gotta take a hike.
(Anyway, that's how I would handle it

)
Good luck!