President Obama was correct when he called Israel a "Jewish State. Israel is a Jewish state it is a state that has been mostly Jewish for a long time. To address in public a persons choice of faith may not be the smartest move but it does not mean that it isnt true.
The United Kingdom maintained Palestine from 1920 until the Jewish People's Council declared the land to a Jewish State, called Israel, in 1948. This new self-declared Jewish State was immediately recognized diplomatically by both sides of the Cold War, the U.S. and the Soviet Union. Soon, massive numbers of Jews relocated to Israel, aided by Israeli law offering citizenship to all Jews. The inclusion of the Star of David centrally in the Flag of Israel reiterates the nation's Jewishness; it is not a relic of several centuries ago, as with religious symbols in the flags of some other countries, but rather was chosen mere decades ago.
The modern Israel was founded in 1948, after World War II and as one response to the murder of six million Jews in Germany by the Nazis. It was founded in the region where the original Israel existed from shortly after the time of Moses up to the era of Jesus Christ. It uses Hebrew as its main language. While there are many secular Jews in Israel, it is, nevertheless, a Jewish State.
No, President Obama was not correct when he called Israel a Jewish State, because there are a lot of different groups of people that live in Israel. There are some Palestinians who live quite peacefully in Israel among all of the Jewish. Israel is actually quite progressively tolerant of other viewpoints.