The term PTO Paid Time off, or sometimes Personal Time Off, is often used for both taking vacations and for sick-time, doctor's appointments, and in other eye problems- as in I just couldn't see coming to work. In fact many companies, in the US at least, officially call it PTO and make no distinction between traditional vacation days and sick days an simply provide a number of days of PTO that may be used either way.
As note though, at other companies I've worked for, the term vacation is used whenever personal time is taken regardless of the reason or duration so it is quite reasonable to hear, "He's taking a vacation day today" as well as, "He's taking PTO today."
As a further note the phrase, "He's off today" certainly means they're not working today, but is ambiguous as to whether it is because they were not scheduled to work or are taking PTO.