Tom Steyer, the billionaire businessman, approached Mr. Sanders in the middle of the exchange.

“I don’t want to get in the middle,” Mr. Steyer said. “I just want to say, ‘Hi, Bernie.’”

Both the Warren and Sanders campaigns declined to comment on Wednesday.

The exchange onstage between the two progressives, surrounded by onlookers but wrapped up in their intensely personal rivalry, marked their most direct confrontation in the entire 2020 election. And in some respects, it represented a kind of inevitable concession to reality: If Ms. Warren and Mr. Sanders share an ideological cause, up to a point, they cannot ultimately share a presidential nomination. For Ms. Warren, taking on Mr. Sanders face-to-face risked further angering the far left, sections of which have already turned on her for the rivalry with Mr. Sanders. And while Ms. Warren was praised for addressing the subject of gender head-on during the debate, even some of her supporters acknowledge that tackling sexism so prominently could risk leaving primary voters uneasy about the implications of nominating a woman.

Progressives fear the public division will benefit the moderates in the race — and, more broadly, threaten the movement they have tried to build. Leading progressive groups spent hours Wednesday trying to craft joint statements of unity while their leading political figures were in a public fight.