Why did the Prophet (peace be upon him) not wash his hand after touching the droppings, and why did he not order the Companion to do so?

It is not established that the Prophet, peace and blessings be upon him, washed his hand from the impure dung, nor that he commanded the Companion to do so. Perhaps the reason for this is the pardoning of dust from impurities, due to the difficulty of guarding against it. For an impure substance does not render impure that which it meets if both are dry.