Invite (all) to the way of thy Lord with wisdom and beautiful preaching; and argue with them in ways that are best and most gracious: for thy Lord knoweth, best who have strayed from His Path, and who receive guidance. (Qur'an 16:125)
January is named after Janus, the Roman god of doors and gateways. He was commonly depicted in statues, carvings and paintings as a two headed man with one head facing forward and the other head facing backwards. In 46BC Julius Caesar chose January 1st as the first day of the New Year as Janus symbolically represented the door to the New Year. Wild parties and orgies were held on the night before the New Year’s Day as a re-enactment of the chaos which Roman mythology depicted as preceding the cosmos or the ordered world whose organization was set by the gods. Furthermore, by that time, Janus had become, in practice, the highest god receiving the ritual sacrifices of Roman worshippers before the other gods, including the chief god, Jupiter.
Thus, in its essence the celebrations of the New Year on January 1st and New Year’s Eve, the night before, are a part and parcel of pagan religious rituals based on idolatrous beliefs in false gods. Consequently, it is completely Haraam (sinful and forbidden) for Muslims to participate in or adopt any of its related rituals, customs and symbols.
If a non-Muslim greets a Muslim, “Happy New Year”, the Muslim is not allowed to respond in a similar manner or say, “Same to you.” Instead, in order not to offend or hurt the feelings of non-Muslim friends or acquaintances, one may say instead, “Happy holiday.”
As for celebrating the New Year according to the Islamic calendar which begins with the month of Muharram, this is also not permissible from a number of perspectives. First and foremost, if one does so believing that it is pleasing to Allah to do so, thereby transforming it into an act of worship, it becomes a Bid‘ahor cursed innovation in the religion about which the Prophet (pbuh) said, “Every innovation in religion is misguidance and all misguidance leads to the Hellfire.” If one does so merely as a custom, it is still impermissible as it falls under the prohibition of imitation of pagan customs about which the Prophet (pbuh) said, “Whoever imitates a people becomes one of them.”
 Pope Gregory 13th who set the modern calendar, the Gregorian calendar, also officially fixed the first day of the year for Christian Europe as January 1st in 1582.