The Lord spoke to Moses, saying, "Speak to the Israelite people thus: In the seventh month, on the first day of the month, you shall observe complete rest, a sacred occasion commemorated with loud blasts. You shall not work at your occupations; and you shall bring an offering by fire to the Lord."
The concept of a "New Year" does not occur in the Bible; even later, the Mishnah's catalogue of "new years" does not highlight any of the various legal cut-off points for tithing etc., as of particular spiritual significance. The first day of the seventh month, in the Bible, just initiates a three week sequence of holy days; the only unique characteristic of the day is the sounding of the shofar. The dramatic pageantry of the High Holy Days in the Temple has been compacted, since the destruction, into the text of the synagogue liturgy.
The dominant Israeli approach to the Jewish tradition has been based on the teachings of Achad Ha'am (Asher Ginzburg) at the beginning of the 20th century (and adopted by Mordecai Kaplan, the founder of Reconstructionism in American Jewish life): Judaism is a culture, not a religion. Our calendar and texts and practices are sanctified not by divine commandment, but by their being central to the "life-force" of the people, by their being the scaffolding of our identity. Achad Ha'am's famous saying about the Shabbat says it all: "More than Israel has kept (observed) the sabbath, the sabbath has kept (preserved) Israel."
And so, over the century of Zionist culture-building, educators and artists and just ordinary folks succeeded in recasting many elements of the Jewish tradition into meaningful secular symbols. The seasonal festivals of Sukkot, Pesach, and Shavuot with their colorful practices and symbols continued to be important milestones in the calendar even for those who never attend synagogue and who don't find spiritual satisfaction in a traditional conception of God or mitzvah. Chanukah took on new meaning related to nationalism and heroism; Purim became a combination of Mardi Gras and Halloween; Tu Beshvat has gone through various stages of development - from a celebration of our attachment to the land and its productivization to a festival of environmental awareness. Even the fifteenth of Av, another non-holiday mentioned in passing in the Talmud, has become a major cultural event, Israel's answer to Valentine's day.
The toughest nuts to crack have been the High Holy Days. How do you secularize a holiday whose only significant observance is to spend many hours in synagogue, praying? How do you find national, cultural significance in a holiday that focuses exclusively on the individual and his/her relationship to God and conscience? The answer to these questions is: with great difficulty, if at all. There have been some attempts to create a home "seder" for Erev Rosh Hashanah, with some readings and songs interspersed with partaking of some of the symbolic foods that are traditional in various ethnic groups. But the majority of non-religious Israelis are at loose ends on Rosh Hashanah. It is a day off, schools and businesses are closed - but what do you do? Many go to the beach, or hiking, it is a 100% occupancy weekend for B&B's and resorts; many go to resorts in Turkey or the Greek Islands.
In recent years there has been an increased interest in the synagogue experience; Reform and Conservative congregations have grown, and now community centers have started running services for the High Holy Days, as many non-religious Israelis would feel too intimidated and alienated to attend the neighborhood Orthodox shul.
It is interesting to consider: is the increased spiritual seeking and experimentation in Israel in the past ten years simply part of a world-wide fad of the post-modern moment in which we live, or is it evidence of the failure of the attempt to redefine Judaism from a religion to secular national culture?