All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
There is no doubt that it is most appropriate to use the Hijri calendar because it is the Hijri year that includes the lunar months which Allah made as measurement of time for the people, for Hajj, fasting, and the sacred months, and it is by these months that the days whose fast is recommended are known. Allah says (what means): {They ask you, [O Muhammad sallallaahu ‘'alayhi wa sallam] about the new moons. Say, “They are measurements of time for the people and for hajj [pilgrimage].”} [Quran 2:189] Moreover, Allah says (what means): {Indeed, the number of months with Allah is twelve [lunar] months in the register of Allah [from] the day He created the heavens and the earth; of these, four are sacred. That is the correct religion.} [Quran 9:36] Also, the Prophet said: ''The year is twelve months, four of which are sacred, three are consecutive: Thul-Qi’dah, Thul-Hijjah and Muharram, and the fourth one is Rajab of Mudhar, that comes between Jumaadah and Shaa’baan.” [Al-Bukhari]
Therefore, it is most appropriate to use the Hijri calendar as we clarified but we do not have evidence prohibiting the use of the Gregorian calendar even though one should not restrict himself to the use of the Gregorian calendar only. Indeed, the Hijri year is the calendar exclusive to the Muslims, and they need it for their festivals, their acts of worship and their dealings – like Hajj, fasting, the waiting period [for the divorced or widowed woman], breastfeeding children, Zakah and so forth.
The Hijri calendar is based purely on the lunar months; Allah says (what means): {They ask you, [O Muhammad] about the new moons. Say, “They are measurements of time for the people and for hajj [pilgrimage].”} [Quran 2:189]. Therefore, a Muslim should not abandon the Hijri calendar and the lunar months to use any other calendars.
Allah knows best.