The biggest strategic fear for Washington, Israel and many Western countries is that the current crisis metastasises into a regional war which spills beyond Israel’s borders.
Iran has already threatened it will not stand by while Palestinians are killed in Gaza.
There are now reports in the Arab media that Iranian-backed militias are moving westwards from Iraq and Syria into southern Lebanon, closer to Israel’s borders.
Some of these fighters have had recent combat experience supporting the Syrian army against rebels.
Of all the various pro-Iranian militias in the region, Hezbollah in Lebanon is the most potent.
It fought an inconclusive war with Israel in 2006 and since then it has restocked its supplies of rockets and missiles that are now thought to number around 150,000.
Some of these are long-range and precision-guided, enough to flatten whole ministries in Jerusalem and Tel Aviv, if they can overwhelm Israel’s air defences.
To deter Hezbollah - and Iran - from joining Hamas’s war against Israel, the US has stationed two entire aircraft carrier strike forces offshore, in the eastern Mediterranean.
Tensions in the region were already high before the deadly hospital blast in Gaza.
If and when Israeli forces make a full-scale incursion into Gaza, there is a serious risk this could be the trigger for Hezbollah to open a new front from Lebanon.
Dousing those flames would not be easy.