I don't believe there is a default keybard shortcut, but you might be able to assign one depending.....
Do want to copy an existing document to a new document or copy the contents of one document to another open existing document?
I don't see how that would work though. You'd need to at least name the target document, so there would not be a one click solution. I just group the layers, drag the source document away from its tab, and drag the group into the new document.
There is of course Image - Duplicate... in the menus.