As for my name: All the girls I know named Noa spell it with no H in English. (We usually spell it in Hebrew, so the English spelling is not that important.) Israelis usually spell their names in English the way they sound, and the H is completely unnecessary.
On the other hand, English speakers do not need to spell names the way Hebrew speakers do. If you spell Yonatan as Jonathan, Noach as Noah, and pronounce Michael as MIE-kull, there is no logical reason to spell Noa the Hebrew way on legitimacy grounds. You might want to do that anyway, so people wouldn't think she's a boy.
And yes, it's extremely popular in Israel. It's been the number 1 name for a few years, with about 3% of the Israeli Jewish girls born each year bearing that name. It's awful.