Is the fast of my sick brother, who wakes up to vomit after the call to dawn prayer, valid, or must he make it up?

Deliberate vomiting invalidates the fast and necessitates making up the missed day. As for involuntary vomiting, it does not invalidate the fast and does not necessitate making it up, based on the Prophet Muhammad's (peace be upon him) saying: "Whoever is overcome by vomiting does not have to make up the day, but whoever vomits intentionally must make it up."