Right, gotcha. Analog inputs in general wouldn't be tied to the digital input of the DAC (this will be true regardless if you're talking about a pair of headphones or any other device), and besides your computer's motherboard wouldn't be able to output a digital signal from the headphone socket*.

(Well, with one caveat: in the past some manufacturers had a combo analog + Toslink connector - Toslink is optical digital connector so your receiving device needs an optical input.)