Malbim Ayelet HaShachar Klal 545
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ושם "חג" הונח על החגיגה שהיו מקריבים בימי מועד; רוצה לומר שלמי שמחה. ועל כן ראש השנה ויום הכפורים לא נקראו בשם "חג" בשום מקום. (אמור ס' קצא)
The noun חג (holiday) is used for the holiday offerings that were sacrificed in the days of the appointed times; it means to say שלמי שמחה (peace-offerings of joy). Thus, New Year’s Day and the Day of Atonement are not called by the name חג anywhere.
On New Year’s Day and the Day of Atonement, God did not command the Israelites to bring up peace-offerings of joy, only on Passover, the Feast of Weeks, and the Feast of Tabernacles. Therefore, the wordחגdoes not pertain to New Year’s Day or the Day of Atonement.