C.D.
Keep it honest, keep it casual. Just give the new friend a call. Tell her how disappointed you are that her child can't come, and explain the circumstances. Ask her when she's available for a playdate, and arrange the plans. While you're at it, put your daughter on the phone and let her apologize for her behavior.
Let go of the guilt and embarrassment. You are not responsible for your daughters words, she is. If you have a good reason for not being able to accomodate another child at the birthday venue, most reasonable people will understand. And don't be to hard on the kid. At this age, they are only beginning to develop the complex emotions required for "social graces." Good luck!