Zelhar is right, does it matter? What matters is that Sharon was an evil man and Hashem has taken him away.
I don't know if he was Jewish or not. From my understanding many years ago the Subbotnik Jews were sincere gerim who were persecuted for their choosing to be Jewish. Today, they are or are not accepted as Jews or have a safek status. Are we going to say that if (A) does a conversion- A's descendants, B, C, D, and E years later are suddenly not Jewish anymore? It is a very hard situation to determine. I think that is where the Chief Rabbinate steps in to determine what Subbotniks are and are not. Some are regarded as Jewish, some are not. As far as I understand there has been no official decree that says that Subbotnik Jews are not Jews. It is more on a case by case basis. If they go to the rabbinate and are told they are not Jews, then they would convert. The problem is, I am not sure how Halachich was the Subbotnik conversion 200 years ago. But let's say that they did do a Halachic conversion and have keen records and have not intermarried.... should we say they're not real Jews? If the Subbotniks did not do a Halachic conversion, then obviously they are not Jews.
Anyways, I highly doubt that Sharon would have cared enough to do a conversion. And even if he had done a conversion he would have been allowed to rule in a position of authority because the country of Israel would follow secular laws. And there are cases that a ger tzeddek who has a Jewish father CAN be in a position of authority. Such as the Chief Rabbi of the Czech Republic. He did a conversion because his mother was not Jewish but his father was. From my understanding, the rules are different for gerim whose fathers are Jewish. This is why this Chief Rabbi was able to get to that position. Had both his parents been non-Jews and he did giyur, then he could not have attained the position of Chief Rabbi:
http://en.wikipedia.org/wiki/Karol_Sidon
Maybe it would apply to a ger tzeddek who would be in government in Israel.