Here’s the difference:
In the United Kingdom, the Queen is the sovereign.
She let ‘em have a vote.
In America, We the People are the sovereign.
Convince a sufficient majority of them to let a part of their country go, and it would be so.
Otherwise, nope.
On the contrary, in the United Kingdom Parliament, not the monarchy, is sovereign, and has been so for a good few centuries. The Queen had nothing to do with 'permitting' the Scottish referendum. The devolved Scottish Parliament decided it would hold a referendum, and the United Kingdom Parliament agreed it would honour the outcome.