I'd do a quick apology, and not overdo it because we've all had kids do things that tick us off. I agree with Suz that the constant repetition when told to stop would be more of a problem, no matter what the issue. It might have made sense to send your son to his room for a few minutes until he could control his mouth (or, if you were out at a venue, then into the car with a seatbelt on until he could calm down). I absolutely agree that he can write a thank you note for the gifts, and should - no matter his behavior, good or bad, he writes a thank you note. If he can't write so well yet, it's okay for him to dictate it to you, and then you write down what he says, no matter how childish (he's a child after all). It needs to be from him, from his heart. My child didn't get to play with gifts until the thank you notes were written.
We read on Mamapedia all the time about adults who have poor manners (no RSVPs, rude comments), so it's time for each of us to take the reins of etiquette and good behavior, as well as pure gratitude and appreciation, and start upgrading our own behavior and requiring it of our children. That might mean actually inconveniencing ourselves, or examining our standards. When we talk up birthday parties and make it about the gifts, we set our kids up for a "gimme gimme" attitude, when it should be about the celebration and being surrounded by friends who are actually happy we were born. If we have huge parties with 50 people, we set the kids up for a "spend-a-lot, get-a-lot in return" dynamic.
So while you are feeling bad and embarrassed, try to switch your focus to the other child who was made to feel bad and unappreciated. Always make it about the other person, and you'll help your child to be a more gracious host and appreciative recipient.
Same thing goes for Christmas or Chanukah or whatever holidays you celebrate - which might be a good next step rather than waiting for his next birthday which is way too far down the road. Instead of making a list of what he wants, make it all about what we are joyfully and happily giving to others. It's not just spending the money - it's choosing what someone will like, and going through the effort of wrapping and delivering, with happiness and not with a begrudging attitude that overtakes us with stress. You could also involve him in some charitable acts - collecting food for the local food pantry or donating outgrown toys and clothes to needy children, and where possible having your son participate in the delivery. It's very eye opening for kids and sets them up for the future.