Without a ceasefire the next most important issue on the agenda of the Biden administration, and which Secretary of State Blinken is reported to discuss, again, with Prime Minister Netanyahu and with a number of regional political leaders, is the avoidance of an "escalation' of the Gaza conflict into a regional war. Such an escalation would present Biden with a difficult political problem at the outset of his 2024 presidential campaign. It would depend on the nature of such an event and the possibility of pressure for the U.S. being drawn directly into it, presumably on the part of Israel. The alternative would be staying out and demanding a hopeless diplomatic solution which would put Biden's credibility as an avowed supporter of a strategic ally versus a conflict avoider based on political pressure from the Progressive Left of his Democrat party. His decision could hurt him in either case in the upcoming election.
But in practical terms the conflict is already a regional war in geographical terms but not much of a "wider war" considering the participants and "relatively" limited nature of the current conflicts and the history of such conflicts. The question is, to what levels the current regional conflicts will expand and who will be the major antagonists.
The prospects for escalation are centered around four areas; Israel's northern border with Lebanon; the occupied West Bank currently under administrative control by the Palestinian Authority and its' Fatah political party; the northern section of Yemen on the tip of the Arabian Peninsula; and U.S. and "allied" bases in Syria and Iraq. The major threat is the border with Lebanon which is controlled by the Iranian sponsored Hezbollah militia. A limited but serious missile attack on Israeli towns close to the border by Hezbollah has been going on for some months. Israel has responded with artillery and air strikes but neither side has demonstrated the probability of ground combat by their respective armed forces beyond aggressive public statements.
Although designated as a "terrorist group" Hezbollah by virtue of its' size, weaponry and combat experience in Syria in support of Syrian President Assad in his battle against a loose coalition of anti-Assad militias, Hezbollah is a formidable adversary. However, Hezbollah does not exist in a social and political environment that is unified in its support for them. The population of Lebanon is roughly one third Christian, one third Shi'a Muslim like Hezbollah, and one third Sunni Muslim. The Lebanese government which is divided along these lines and maintains a national military, is conflict averse and opposes any expansion of Gaza war into its' territory. Memories of the fifteen year long civil war (1975-1990) and its' economic and structural destruction as well as the deaths of 120,000-150,000 people, are still strong. This was followed by the short Israeli/Hezbollah war in 2006. A Hezbollah missile attack on major Israeli cities or a major cross border attack would cause missile sites and bases in Lebanon to come under attack as well as a response by Israeli ground forces. Clearly, the threat of "regional expansion" of the Gaza conflict lies in the hands of Hezbollah and its' sponsor Iran, not with Israel whose leaders have made it clear that they seek no large scale conflict in Lebanon and who have exercised considerable restraint in the face of missile attacks on border communities. But those attacks have resulted in the evacuation of 150,000 Israel citizens from their homes creating a situation which cannot be tolerated indefinitely.
With respect to threats to widen the Gaza war on the part of the Houthi militias in Yemen, the threat they represent hardly deserves their inclusion. The Houthis are another Shi'a tribal surrogate of Iran. They are themselves operating under a truce with government supporters suspending a lengthy civil war (2014-2022) which wrecked havoc with Yemen's civilian population and remains politically unsettled. Saudi Arabia, which shares a border with Yemen, supports the internationally recognized government opposing the Houthis and supported that government militarily during the conflict. A widening conflict between the Houthis and the U.S. led international maritime presence in the Red Sea could upset the truce in the Yemen civil war which would not affect the Gaza war but put the Houthi resistance movement at significant risk.
Yemen is the poorest nation in the Arabian peninsula. It's only significant sources of export income are degrading oil and natural gas reserves. Yemen itself is 1,600 miles from Israel and the Houthis have expressed their "support" for Hamas by attacking international shipping in the Red Sea in transit to and from the Suez Canal. They have done this with armed drones, missiles, and small boats. This in no way helps Hamas or hurts the Israeli war effort, and after questionable delays the Biden administration has organized an international naval response to protect civilian shipping and has "implied" that further attacks will bring a more serious military response, presumably, a further much delayed attack on the Houthi missile and small boat launching sites.
The Houthis have tried to use their "support" for Hamas and the Palestinians as a tool to generate domestic support in the face of severe economic hardship in the territory they control but time is running out for their interruption of vital international maritime trade.
Other concerns about a "regional war" include attacks by a variety of small Islamic terrorist groups on American and allied troops in Iraq and Syria. The number of U.S. military personnel is relative small at around 2,500 and consists of "advisors" who are part of a continuing operation against the international terrorist Islamic State. These attacks and counter attacks have been small so far without American fatalities and the idea that Biden will undertake a major military operation as a result is unrealistic. The easy solution would be to withdraw this small military presence which is not going to be a deciding factor in the local terrorist problem in any case. There presence does not have the full support of the Iraqi government in any case.
The fourth concern about a widening of the Gaza war most mentioned is a dramatic increase in anti-Israeli terrorist activity or even a third Intifada, or popular uprising, by Palestinians, in the West Bank. So far the increase in hostilities there have been mostly on the part of radicalized Israeli settlers who have been attacking Palestinian civilians. The Israeli military who have been attacking small Palestinian terror cells in the West Bank must intervene to stop the radical settlers from further inflaming the local population and should cooperate with the Palestinian Authority administrators to the extent they can, to limit the expansion of hostilities. In any case the situation in the West Bank is not a part of a "wider regional war" although it could complicate the Israeli war effort in Gaza. There is no heavily armed or organized Palestinian resistance in the West Bank and tensions and conflict have been part of the Israeli occupation for decades. While Iran could conceivably attempt to intervene indirectly by smuggling arms into the West Bank, it is not in the interests of the Palestinian Authority which exercises domestic governance and receives significant monetary aid from abroad, to assist or participate in a general uprising. In most of the post-Gaza war discussions so far the Palestinian Authority is mentioned as serving as the fundamental governmental entity in a future Palestinian state which would include Gaza. Officials in the PA will not want to put that scenario at risk by taking an active role in opening a new anti-Israeli insurgency.
Is a wider regional war possible? Of course, wars are essentially unpredictable. But Israel is not seeking such and escalation as it devotes it attention and resources to the conflict with Hamas in Gaza. A wider war is largely up to Iran whose regional aspirations are behind its sponsorship of Islamic client groups. A direct conflict with Iran is highly improbable as Iran has demonstrated an unwillingness to expose itself the destruction such a conflict would bring and it prefers a strategy of attrition towards Israel through the use of its' surrogates and now the domestic political/electoral pressures on Biden's Israeli policy of "rock hard support".
The war in Gaza will end when Israeli leaders accomplish their goal of eradicating the Hamas leadership and disarming the rank and file; or when the Hamas leaders see the end in sight and flee to a sympathetic nation, although the list such places is quite short. Of course the subsidiary conflicts in the region will continue as they have for decades and it will take decades more for a permanent solution to the Palestinian problem, if one is even possible.