Space entrepreneur Elon Musk has stoked excitement about accelerated development of private deep-space vehicles—while provoking consternation inside NASA—by projecting his proposed megarocket targeting Mars could take humans on an orbital test flight around Earth as soon as next year.

Mr. Musk’s announcement over the weekend sketched out a superambitious timetable for building and testing a privately funded, behemoth rocket intended to take people to the moon and eventually to Mars. But even before details emerged, the head...