As American cities face a surge of coronavirus patients who will require lifesaving care, they are also facing a rapidly dwindling supply of available intensive-care-unit beds. Physicians and nurses at many of the country’s largest hospital systems are leveraging a decades-old technology in new ways—at times coupling it with cheap, readily available gadgets—to expand their ability to care for and monitor patients.

Generally known as “tele-ICU,” this two-way bedside video is sort of like FaceTime or Zoom. The difference is...