Updated at 6:35 p.m. EDT

While President Biden’s characterization of an Israeli roadmap to ending the war may fail to free hostages or lead to a Gaza cease-fire, it is succeeding in deepening Israeli political enmities and strengthening Hamas’s prospects of survival.

In a speech Friday, Mr. Biden seemed to publicize a plan that Israel had hoped to keep under wraps. His aim may have been to force the players’ hands while allowing enough ambiguity so each side could declare victory — yet shall the twain ever meet?

Late Monday, the American ambassador at the United Nations, Linda Thomas-Greenfield, circulated to members of the Security Council a proposal for a resolution based on Mr. Biden’s speech. “Ultimately, it would lead to an end of the war in a manner that ensures Israel’s security and brings immediate relief to the civilians of Gaza,” she wrote in a statement. “The Security Council must insist that Hamas accept the deal.” 

But can it? Hamas’s declared demand for the release of hostages, from which it has never retreated, is that Israel permanently end the war and retreat from Gaza. Israel, on the other hand, refuses to stop the war before Hamas is dismantled as a military and governing force. 

“Israel is willing to allow Gazans to return to the north of the strip, it is willing to release terrorists with blood on their hands,” a Channel 12 analyst, Amit Segal, says. “But ending the war is unacceptable.” That position, he adds, is shared by two opposition leaders, Benny Gantz and Gadi Eisenkot of the National Unity party, who have joined Prime Minister Natnayhu’s war cabinet after October 7. 

Since Friday, Hamas has kept mum about Mr. Biden’s speech, other than a vague, noncommittal reply made by a Cairo official. “Hamas’s initial statements indicate that it received the proposal positively,” Egypt’s foreign minister, Sameh Shoukry, said Monday. 

Israel, meanwhile, is torn apart. Under pressure from all sides, Mr. Netnayahu says Mr. Biden’s speech was an “inaccurate” portrayal of the Israeli proposal. The premier’s far-right coalition partners, Itamar Ben Gvir and Betzalel Smotrich, are threatening to bolt his ruling coalition, while left-wing politicians and family members of the hostages are protesting and demanding the deal be embraced.  

In fact, though, even hostage families are torn over the deal. “My father is supportive, and it’s understandable, but this scary deal would not bring my brother back,” Nadav Miran told Kan news. His brother, Omri, was featured in a recent Hamas hostage video.

The Israel Defense Force announced Monday that four men who were held in Gaza and were assumed alive have been killed by Hamas. 

Israel must exert much more pressure on Hamas, Nadav Miran says, adding that to date negotiations have been wrongheaded. “We need to bring all of them home together, not in a drip-drip manner,” Mr. Miran says. “And we need to do it fast, because more and more of them are killed each day.”

The four-and-a-half-page Israeli proposal, in contrast, entails a phased hostage release. In the first 42-day stage, as Mr. Biden detailed it, there would be a “full and complete cease-fire.” Israeli forces would withdraw from densely populated areas, and hundreds of terrorists held in Israeli prisons would be released in exchange for an undetermined small number of women, elderly, and wounded hostages. 

In the third phase, billions of dollars would be invested in Gaza reconstruction. Yet, Jerusalem and Washington disagree over the details of the second phase, in which all remaining hostages, dead and alive, must be released. 

If Hamas “lives up to its commitments, the temporary cease-fire would become, in the words of the Israeli proposal, ‘the cessation of hostilities permanently,’” Mr. Biden said Friday. On Monday, though, Mr. Netnayhu told a closed-door meeting of the Knesset’s foreign and defense committee that Mr. Biden’s presentation was “incomplete.” 

“I think of the hostages all the time,” Mr. Netanyahu later said in a videotaped Hebrew-language statement. Yet, “we also maintain the other goals of the war, and chief among them is the dismantling of Hamas.” Both goals are included in our roadmap, he said. “This is not something I’m now adding because of pressure from my ruling coalition. It is what we agreed on in the war cabinet.”    

Mr. Netanyahu spoke after a tense meeting with Mr. Ben Gvir, who said afterward that the proposed roadmap, which was chartered without his consent, needs the approval of the larger cabinet to become legal. Mr. Netnayhu is reneging on his promise to share the full text of the proposal with him, Mr. Ben Gvir said.   

Mr. Gantz said last month that he would leave the alliance with Mr. Netanyahu by June 8. Now, though, he might stay to secure the proposed deal. A fierce Netanyahu opponent, Yair Lapid, says that his opposition Yesh Atid party would vote to keep the Netanyahu government in power to ensure that the deal is implemented.      

In Washington, the National Security Council’s spokesman, John Kirby, insisted that there is no difference between the Israeli roadmap and Mr. Biden’s Friday speech. The president made it public “to try to energize the process” and force Hamas’s hand, rather than pressure Israel, Mr. Kirby said.

Hamas, though, is unlikely to release hostages while it is watching a growing division among Israelis, and as Mr. Netanyahu struggles to keep his coalition intact.