There is nothing that prevents an Arab, a Chinese person or a Martian for that matter. The requirements are that the Prime Minister of Israel must be an elected member of Knesset (parliament) and he must be over the age of 21. There are different rules over who is allowed to run for parliament, they include Israeli citizenship and the requirement of not holding various public or military offices. In order to get enough votes, the party must have a platform that will attract voters. The party that is most likely to form a majority or coalition majority is requested by the President to assemble a government. If no Arab members of Knesset are asked to form a government, it is because they are unable to present a platform that Israeli citizens will vote for, just like in any parliamentary democracy.
U.S. warns governments, bankers Iran sanctions still in place (Reuters)