Here is the Daf Yomi from Mesechet Shabbat which discusses this issue you raise t_h_j:
http://www.shemayisrael.com/dafyomi2/shabbos/insites/sh-dt-129.htm3) USING GENTILE DAYS AND DATES
In the previous insight, we showed that the days of the week come from the names of gods in Greek and Norse mythology. This raises the question of whether it is at all permissible to refer to the days of the week by such names. This also raises the broader question of whether we may refer to the months of the year by their Julian names, some of which are named after Roman gods (such as January, from Janus, known as the god of the doorway).
Similarly, is it permitted to refer to the year by the number used by the Gregorian calendar, which refers to the death of the god of the Nazarenes? Furthermore, is it permitted to refer to the hours of the day (e.g. 6:00 in the morning), which begin their count from midnight, which stems from a system of belief that maintains that their god was born at midnight?
(1) MONTHS AND DAYS
QUESTION: There are two reasons to prohibit using the gentile names of months and days:
(a) The MAHARAM SHIK (#117) writes that one should not use the gentile names of months, and certainly not the gentile count of the months (putting January as the first month). The reason is because their system does not make Nisan the first month, and we have a Mitzvas Aseh to refer to the months from Nisan, counting Nisan as the first month, in order to always remember the redemption from Egypt (RAMBAN, Shemos 12:1). The similar logic applies to the days of the week. The Gemara (Beitzah 16a) says that the days of the week should be referred to in relation to Shabbos ("the first day from Shabbos" and "the second day from Shabbos", etc.) in order to give honor to Shabbos. It should, therefore, be prohibited to use the gentile names of the days of the week.
(b) The names of the months and the days of the week are based on names of gods that were used in idol worship. Are we permitted to use such names?
(a) In response to these problems, the Ramban in Parshas Bo writes that with regard to the months, when the Jewish people returned from Bavel to the land of Israel, they named the months by Babylonian names (which are the names that we now use) in order to remember the redemption from Bavel (see TOSFOS, Rosh Hashanah 7a), the same way that until then, they referred to the months in relation to Nisan in order to remember the redemption from Egypt.
The SEFER HA'IKRIM (3:16) understands this to mean that when the Jews were exiled to Bavel, effectively bringing an end to the liberty they had enjoyed as a result of the redemption from Egypt which occurred 890 years earlier, there was no longer a necessity to count from Nisan to recall the redemption from Egypt (see also CHASAM SOFER, Choshen Mishpat 1, DH Nachzir).
However, the PERUSH HA'KOSEV in the Ein Yakov at the beginning of Megilah (3a) strongly opposes this view and explains that when the Jews left Bavel they only *added names* to the months, but they did not change the numbering system; they continued to count the months from Nisan. It is permitted to refer to each month by its name, but when one gives each month a number, one must count the month based on the original system, with Nisan as the first. This opinion is supported by the GET PASHUT 127:35, MINCHAS CHINUCH 311:3, and RAV OVADIAH YOSEF in YABIA OMER 6:9:4.
Therefore, one should refrain from referring to the months by the gentile numbering system (e.g. referring to January as "1"). (It should be noted that the months of September, October, November, and December are named according to their numbers ("septem" = seven, "octo" = eight, "novem" = nine, and "decem" = ten). Interestingly, these numbers are not in reference to January, since two months were added at a later point in time. It so happens that they conform to the count from the time of the year which usually corresponds to Nisan!)
For the same reason, as far as the days of the week are concerned, it seems that one who uses their names and not their numbers does not transgress a Mitzvas Aseh. However, it may be prohibited to refer to the days of the week by a different *numbering* system (for example, calling Monday the first day of the week).
(b) With regard to mentioning the names of idols, since these idols are no longer known or worshipped in the civilized world, it should not be prohibited to mention their names, since one has no intention to refer to those idols when he says the name of the day or month.