I am a one weekend a month ARNG soldier. My wife is supposed to give birth a few days before my next dril. Is there a reg. that excuses me from drill for a time being after the birth of my child?

None specific to child birth for dad. All drill absences are regulated by AR 135-91. Call your chain of command, let them know what's going on. they may let you come in during the week and make up the drill. Above all don't just be a no-show/no-contact; then they'll have no interest in allowing you to make up the drill.