"If a prime minister can't lead, she should leave," the Labour leader told MPs in the House of Commons.
Last week, May fought off a plot to oust her by around 30 of her Conservative lawmakers, sparked by her disastrous speech to the party's annual conference.
But her position remains fragile, even as negotiations on Britain's withdrawal from the European Union approach a crucial stage.
In the first prime minister's question time since the plot was revealed, Corbyn focused on pressing May on welfare reforms, which he said were leading claimants into poverty.
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"This government is more interested in fighting amongst themselves than in solving these problems."
May responded by listing the government's achievements on the economy, jobs, health and education -- while accusing Labour of struggling with anti-Semitism in its ranks.
Conservative MPs also heckled Corbyn when he spoke of in-fighting, referencing the fact that only a year ago, his own lawmakers had tried unsuccessfully to remove him.
May's authority was severely weakened by a snap election in June that she called to cement her power, but which saw the Conservatives lose their parliamentary majority.
Johnson and other senior ministers rallied around at the weekend, but questions remain over how long May can last.
The open divisions come at a crucial time in the Brexit talks, as EU leaders next week are due to decide whether the negotiations can move past the terms of the divorce on to the issue of trade.