There is a lot of discussion concerning why Muslims do not celebrate Christmas, even though they accept Jesus as their Messiah, and this more so deals with the paganistic aspects of the day than anything to do with Jesus himself. Even scholars of Christianity acknowledge that the 25th of December was selected for liturgical purposes, as a means to subsume the important Roman holiday of Dies Natalis Solis Invicti (birthday of the Invincible Sun [God]).
The reality is that the Islamic tradition not only celebrates the birth of Jesus, but celebrates it on a weekly basis. In Usul al-Kafi we find a narration from Imam Musa al-Kazim, in which he states:
وأما اليوم الذي حملت فيه مريم فهو يوم الجمعة للزوال وهو اليوم الذي هبط فيه الروح الأمين وليس للمسلمين عيد كان أولى منه. عظمه الله تبارك وتعالى وعظمه محمد (صلى الله عليه وآله)، فأمر أن يجعله عيدا فهو يوم الجمعة
“As for the day on which Mary conceived, it was Friday at noon, and that was the day that the Ruh al-Amin (the Holy spirit) came down, and there is no festival (Eid) better than this for Muslims. Allah, the Blessed and Almighty, magnified it, and Muhammad (S) magnified it and He ordered that it should be a holiday, and it was Friday.”
This is just one of the many reasons why Muslims view Friday as their most holy day of the week. Regarding the actual date of the birth of Jesus, we find in Man La Yahduruhu al-Faqih a narration from Imam ‘Ali al-Rida:
ليلة خمسة وعشرين من ذي القعدة ولد فيها إبراهيم عليه السلام، وولد فيها عيسى بن مريم عليه السلام وفيها دحيت الأرض من تحت الكعبة فمن صام ذلك اليوم كان كمن صام ستين شهرا.
“On the night of the twenty-fifth of Dhu al-Qa‘dah (11th Islamic month), Abraham was born and [during the day] Jesus the son of Mary was born, and on that day the earth spread out from beneath the Ka‘bah. Whoever fasts on that day, it is as if he were to have fasted for sixty months.”
Therefore, Muslims do celebrate the birth of Jesus, albeit not on the 25th of December. They view his virgin birth as special, and they observe thanks to the Almighty by praying and fasting.