C.R.
We have all been there done that!
I feel that I need to teach my kids what behavior is allowed, especially in public. If my child throws a tantrum in a store then I give him the opportunity to play it out, for a minute or two. If it is one of those fits that will last longer, I do leave the store but I don't go home. I take him to the car, buckle him in his car seat and let him scream it out. He is safe and we are out of the store. This doesn't stop the staring of people passing by, but he does usually stop faster. Once he is calm, I go back in and finish my shopping. I don't give him the option of throwing a fit and ending my day of errands.
Just remain calm, and don't worry about people staring. If they are parents they know exactly what you are going through, if they aren't parents they won't understand no matter what.
I wanted to add to make sure to plan trips to the store during time he isn't sleepy or hungry. If I am going to be at the store or running errands for a while I take a small toy, a sippy cup and a snack. It makes life so much easier. I have a snack holder that I love called OnTray 2 Go, www.ontray2go.com
Good luck, he will out grow these.