I've been reading a number of "right-wing" blogs that are trying to delegitimacize the current government in Israel. The basic idea is that since the public voted for the Likud and not Kadima and kadima only has 13 knesset members then Kadima had no right to take the prime ministership (and government ministeries...) with them. The ideas are based around the fact that the current government sucks, is a minority, and is going against the ideas of the people who voted for them. People also feel that the Likud should have retained the prime ministership after Sharon dropped out instead of going to Olmert. These are nice and interesting ideas and reflect the ignorance that people have of the political process.
Even though the government might suck, it is quite legitimate. As we've mentioned before democracy does not equal morality. In a democratic Palestinian society, where the government votes to destroy Israel (or even in an Athenian democracy with a referendum of all the people) , that would not nullify their status as a democracy. It would mean that they are bad people who vote for bad things. We should then hope that our elected representatives order our army to wipe them out first.
What they're missing here is that if any other person in the Knesset could have gotten 60 MKs to support him then he can take over the prime ministership. The leadership of the country has less to do with who's in charge of any specific party as much as who can pull a majority together.
As such, after the public voted for their favorite parties to be MKs the MKs then vote on who becomes prime minister. A no confidence vote win is when 61 or more MKs vote that they do not support the government anymore. At that point either the government falls and there are new elections within 90 days or someone else can come up with 61 or more supporters and the President will appoint him prime minister.
Nobody voted for the Likud to head the government. Knesset members voted for Sharon to head the government. Sharon appointed Olmert as his replacement, as stipulated by Israeli law. He could have appointed anybody he wanted from any party, including Shimon Peres.
If another member of the Likud had the support of 61 MKs and the leader of the Likud didn't, then that person, not the party, would have taken over the government. If a party with 2 MKs (I think that's the minimum) could make enough deals to get enough support, then they would be prime minister.
The Likud tried to make a deal with Labor offering the prime ministership to Amir Peretz, but he refused to play ball (a dumb idea from his perspective, in my opinion, but probably very good for Israel as a whole and society at large).