Yes, both are correct. The first doesn't convey as much information. Tied it on someone? Or tied it to something? The 'on' in the second version implies that it is onto someone, usually oneself. That version would be the most common one to use when referring to a scarf.

Both are correct, but the meaning is different. 'She tied the scarf' means that she tied a bow or a knot in the scarf. 'She tied the scarf on' means that she attached the scarf to someone or something by means of a bow or knot.