Good question. So why weren't they allowed to secede? Do I think they had the legal right to secede? Yes. (Tabling for now the WHY for secession, this is about COULD they secede.)
So why didn't the U.S. allow the southern states to secede? I dunno. I do know that, just 85 years before that in 1776, England didn't let the U.S. secede without having to fight for it. Nor did Spain let Mexico secede in 1821. Nor did the new nation of Mexico let Texas secede in 1836. Nor did the U.S. allow the Confederate States secede beginning in late 1860.
I'm not saying it's right. I'm just saying it was the norm for this area. At least as far as I can tell. So maybe nobody should be surprised.
Money. The Southern states were pumping 700 million through the Northern economy each year, and they were producing between 72% and 78% of all the taxes used to run the government.
Secession would cost the North millions as the South took away much of their European trade, and stopped paying the tariffs that kept the US government running.