Resignations of national leaders take different forms.
At 9:00 pm, Eastern Time on August 8, 1974, Richard Nixon, who had been accused of wrongdoing, announced his intention to resign the office of president of the United States. The next morning, he was carried away from the White House by helicopter. By noon on August 9, his resignation had become effective, as he was flying over the midwest.
On July 30, 2008, Ehud Olmert, who had been accused of wrongdoing, announced his intention to resign the office of prime minister of Israel. More than six months later, he is still on the job, with no set date for his departure.
An American president has a vice president (or others, if that office is vacant) ready to take over immediately, if necessary. In Nixon's case at that time, that person was, of course, Gerald Ford. Replacing an Israeli prime minister seems to be more complicated.
Before it's over, the Israeli transition will involve four major steps, three of which have been completed: 1) the selection of a new leader for Olmert's Kadima Party, 2) that leader's (Tzipi Livni's) attempt to form a new coalition (which failed), and 3) a general election. Each of those steps took a matter of months.
Now, a new set of coalition negotiations must begin, and it's expected to take a few weeks. Olmert will remain as caretaker prime minister during that period.
As I wrote here, a distinctive political culture evolves in each country. So, I'd be very reluctant to suggest that Israel should follow our way, or vice versa. But I find the difference interesting.
The rest of the world is primarily concerned about whether an Israeli political deadlock will further delay a resumption of peace negotiations. Until recently, they could not have resumed anyway, due to the American presidential transition. But now that that's settled, attention is focused on Israel's transition.
Aside from the failed experiment I described here, the Israeli electoral system has pretty much stayed unchanged in its six decades of existence. Now, there is talk of another reform effort, this time focusing on reducing or eliminating the degree of proportional representation.