4 week old "overeating"???

I have a 4 week old son who is exclusively breastfed. He loves to eat and does so often. 
However, in the past week or so he has started spitting up large amounts of milk. It almost looks like he's vomiting because it just comes gushing out. He doesn't do it after every single feeding or even every day, it just happens randomly every now and then. 
He is gaining weight and has many wet and dirty diapers. Our pediatrician says he is overeating and that's why he is spitting up so much. He told me to only let him eat every 3 hours. I didn't even know it was possible for a breastfed baby to overeat! 
My baby hates pacifiers and screams horribly when I try to give him one. I try to bounce him or rock him or anything else to soothe him but all he wants is to be nursed. I feel awful watching him cry and smack his lips and not breastfeeding him. 
Do you think my pediatrician is right or should I continue feeding whenever my baby wants to? If you do think I should limit to every 3 hours, is there anything else I can do to help keep him happy when he is wanting to eat but can't?