Do arduino librarys require the arduino to be connected to a computer?
No.  Arduino libraries (strictly speaking only the utilized parts) are permanently built-in parts of your compiled sketch.  Effectively they are "statically linked" unlike the situation with modern RAM-based computer operating systems where compiled dynamic link library code may be separately loaded into memory for a program to call at runtime.  

(This isn't even really a design choice, as the basic AVR Arduino's cannot load new program code into executable memory at runtime anyway - except by reflashing themselves like the bootloader does, which would be a bit slow compared to normal dynamic linking)

Of course if you have a library which is intended to perform communication with a computer, then that connection would probably be required for it to fully operate.