Professor Warrick offers the following initial thoughts:
"In regard to the failed Egyptian ceasefire proposal - I think there are several relevant factors here. First, the proposal came from Egypt, and the Egyptian government and Hamas are not on friendly terms, to say the least (Hamas has been implicated in Egypt's security problems in the Sinai, among other things, as Hamas is a Muslim Brotherhood affiliate, and Egypt's current regime is very much at odds with the MB domestically). Also, Hamas' position is that they were never consulted about the ceasefire or what terms would make it acceptable to them, so they saw no reason to agree to it because it didn't accomplish anything on their agenda (their stated goals being to stop Israeli attacks but also to end the effective Israeli blockade and security controls that have made life very difficult in Gaza).
Israel, on the other hand, briefly accepted the proposal but then withdrew that in the face of multiple rocket launches from Gaza. Some commentators have suggested there's no downside to Israel in accepting a cease fire - either it works, and the rockets stop, or it doesn't work and they can claim justification for continuing military action against Gaza. The ceasefire agreement was definitely unpopular on the Israeli far right though, it's worth noting.
This is unlikely to be resolved unless Hamas and Israel can agree important concessions at the same time: Hamas to stop rocket fire (which is tougher than it sounds, because quite possibly not all of these rockets are being fired by Hamas itself) and Israel to ameliorate their controls on the Gaza border, which affect much more than just military materiel. There have been agreed ceasefires before (Hamas accuses Israel of having broken the last one) but it's not clear (to me at least) that the US is actually a useful party in brokering a new one, since we're a pretty one-sided actor in this. But the US could work more seriously on humanitarian conditions in Gaza, which would mean publicly and credibly pressing Israel to alter some policies that would make a difference there.
There is not much domestic pressure for the US to act, though. Unless Gaza decides to play for Cleveland next year, most Americans just aren't going to pay much attention to this round of conflict, I'm afraid."