There is no such thing as an "Islamic" vs "non-Islamic" or "Muslim" vs "Christian" calendar.
The Qur'an clearly spells out that the sun and the moon, both, are God's creation and both can be used to calculate time and make our calendars. Doesn't it give us a beautiful diversity? See the verses below.
6:96 He is the Cleaver of the daybreak. He has appointed the night for stillness, (rest and tranquility) and the sun and the moon to run their well-calculated courses. They become means of calculating time for you (10:5). Such is the Design of the All-Powerful, the Knower.
10:5 He is the One Who appointed the sun as a splendid glow, and the moon as a cool light. He appointed stages for the moon, in due measure. You make your calculations and calendars accordingly. God has created the Universe as the absolute reality and for a purpose. God explains His verses for those who wish to learn.
24:44 It is God Who alternates the night and the day. Herein is a lesson for men and women of vision and understanding. [Of darkness of ignorance and light of knowledge, of death and life, calculating time, rest and work, planning the life on the planet, and so on]
55:5 The sun and the moon run by a mathematical design. [Such are the changeless Divine laws]
Wassalam,
SA
SA