Within hours of the death of the Hamas leader Yahya Sinwar on Thursday, President Biden and his aides scrambled to design one last push for a broad de-escalation of violence in the Middle East: a cease-fire and hostage deal in Gaza, a pullback from Israeli attacks on Hezbollah in Lebanon, a confident declaration of victory by Israel that might allow it to forgo a major retaliation against Iran.

“It is time for this war to end,” Mr. Biden said as he emerged from Air Force One when it landed outside Berlin late Thursday. He added that he had called Prime Minister Benjamin Netanyahu of Israel and urged him to “move on” from the war and focus on building a new political landscape for the region.

Vice President Kamala Harris, carefully navigating the campaign politics of the moment, declared “it is time for the day after to begin” — a phrase suggesting that after three weeks in which Israel eliminated the leaders of Hezbollah and Hamas, its goals to defeat its adversaries had been met.

Mr. Netanyahu delivered the opposite message: “This war is not over.”

The stark difference encapsulated the argument that has been the consistent theme of Mr. Biden’s often angry exchanges with Mr. Netanyahu for the past year. Now, with only three months left in office, closing that huge gap will be a final diplomatic mission of his presidency.

Nothing in his face-offs with Mr. Netanyahu suggests that the Israeli prime minister will take his advice or seize the chance to turn the military victories into a lasting political accomplishment. One of Mr. Biden’s senior aides said the administration’s concern was that the killing of Mr. Sinwar, and before him the killing of the Hezbollah leader Hassan Nasrallah, ratifies in Mr. Netanyahu’s mind his certainty that he was right to deflect American calls for de-escalation over the past few months.

But this time, more out of hope and exhaustion than evidence, administration officials suggest things might be different.