I’m all for holidays in general, and I’ve heard important rabbis say that Rosh Hashana is indeed a proper time for all humanity to take a “time-out” to examine their lives. We are taught by our Sages, “On Rosh Hashana all human beings pass before Him like young sheep, as it is said (Ps. 33:15): ‘He fashions all their hearts together, Who understands all their deeds’.” Therefore, one might think it to be an appropriate national holiday, aside from any possible Constitutional issue.
However, I really don’t think that Rosh Hashana should be a national holiday in the United States for an important practical reason. When the Jewish children get out of school on Rosh Hashana, while all their friends are in school, it makes the Jewish children realize that they are special. The same can be said for Jews in a non-Jewish work environment - observing Rosh Hashana distinguishes them as Jews. In our age of rampant assimilation and intermarriage, making Rosh Hashana a national holiday might further weaken Jewish identity, removing what for many is the only practice that sets them apart as Jews.
In addition, it is clear that Rosh Hashana is a purely “Jewish Holiday” in the sense of the mitzvah of Rosh Hashana to hear the shofar and celebrate it as a “Yom Tov”.
On a personal note, I grew up in just on the border of two counties – one with a predominant Jewish population and the other without. The schools in my county were open on Rosh Hashana as the schools on the other side of the street were closed. I didn’t go to school on Rosh Hashana, like other Jewish kids in my neighborhood, but I needed to bring an “absence note” from my parents after the Holiday. I still recall as a shy first-grader handing the note to the teacher, waiting to be told how much I had missed – but instead the teacher told me that he wishes that our county would be like the one next to ours and be closed on Rosh Hashana!