Sagamihara Central Hospital said some staff took time off after losing access to child care.
Japan has yet to reach the mass infections seen in other developed nations, with just under 12,000 cases as of late Thursday.
He said a caller from a parent-teacher association also asked the hospital to tell staff to keep their children away from school.
One nurse at the Gifu hospital was blocked from entering a classroom when she dropped her two children off at a day care center.
Unlike the U.S. and some countries in Europe, loud public outpourings of gratitude to medical professionals are nonexistent in Japan.