Obviously, it seems silly for him to plead the 5th on such a question, which is why Hawley asked it, to demonstrate how unreasonable Fauci was being.
Devil's advocate (that is, advocate for Fauci): If you start answering questions, you start down a slippery slope, and you might get trapped into answering something you don't want to answer. Easier to just not answer anything, and let them put you in contempt for not stating the day of the week.