First of all, stop being embarrassed. All kids do this at some point. Sometimes they are 2, sometimes 5, sometimes 14! If you worry about appearances, you stop thinking clearly about how to handle it.
I think that, at this age, canceling a park date for tomorrow based on what happened today may not be effective. Instead, have a sit-down about the rules. Let him know that, if he has a tantrum, he's going home immediately. If you stop him in his tracks and put him right into the car seat (and yes, that means carrying him or dragging him back to the car with people watching), then he'll figure out that consequences are immediate and as promised. Don't worry about the others - they've all seen it, or will soon, in their own lives. If you're with someone, don't go crazy with shame and apologies, but just say "Joey is engaging in unacceptable behavior so we need to leave now. Hope we can get together another time. Bye."
So no you can't keep him home, and it wouldn't work anyway. A 6-week grounding is appropriate for a 14 year old but not a 4.5 year old. And he may not snap out of it - you may have to teach him to come out of it.
If he has a tantrum at home, just say "It sounds like you are really frustrated. If you can calm down, you can tell me what it is. If you can't calm down, you can take some time in your room until you feel better and can stop screaming." If he's blaming others and it's obviously not true, you can stop that by putting him in his room until he stops lying or fibbing.
If he's sleeping so much, you're right, he could be going through a growth spurt, or maybe he's fighting off something and he just doesn't feel like himself. He's frustrated, he can't express it, so he acts out. Sometimes, when a child is calm (either on a good day or when he comes out of his room after being sent there), you can sit him down and try giving him some choices of vocabulary to express himself. Helping him with his words (while being careful not to give him excuses) can give you a better idea of what's going on. But the main thing is, you'll teach him that words are the best way to express his frustration and that you allow and encourage it, but that engaging in tantrums or blaming or disobedience will always ALWAYS result in him having a whole lot less fun.