Throughout time, Yisrael has worked on Rosh Chodesh, being that because of the of the sin of the golden calf the privilege of not working was then stripped from the men. However, some women still hold this tradition. Within Ezekiel 46:1 it says, “‘Thus said the Master יהוה, ‘The gate of the inner courtyard facing east is shut the six days of work, but on the Sabbath it is opened, and on the day of the New Moon it is opened.” With that being said, because the gates were open on Shabbath and Rosh Chodesh, but shut during the six working days, is Rosh Chodesh a part of the six working days?
You ask whether Rosh Chodesh is a part of the six working days.
The Radak and the Metzudas Dovid on the possuk in Ezekiel 46:1 that you quote say that the eastern gate to the Ezras Yisroel was closed on working days because neither the people nor the Nosi came to the Temple except on Shabbosos and Roshei Chodoshim.
The Chomat Anakh (by the Chida) on the other hand says:
שער החצר וכו' ששת ימי המעשה וביום השבת יפתח וביום החדש יפתח. פשטיה דקרא דבר"ח אין עושין מלאכה אך אפשר שאינו מן הדין אלא מנהג והכי משמע מקרא דשמואל דכתיב אשר נסתרת שם ביום המעשה ותרגם יונתן ביומא דחולא משמע דגם האנשים לא היו עושים מלאכה בר"ח. אמנם רוב הראשונים כתבו דדוקא הנשים הוא שאינם עושות מלאכה. וכ'
The simple meaning of the possuk is that on Rosh Chodesh work is not done. But it is possible that this is not a din but a minhag. It is evident from Targum Yonoson on a possuk in 1Shmuel 20, 19 that men also did not work on Rosh Chodesh. But most Rishonim write that only the women were the ones that didn't work.
From Radak and the Metzudas Dovid, it appears that Roshei Chodoshim are different to working days in respect of attendance at the Temple. No implication about working on Rosh Chodesh can be deduced.
The Chomat Anakh points out that men may not have worked on Rosh Chodesh not as din but as minhag and most Rishonim agree only that the women were the ones that didn't work.