Does Christianity allow Christians to celebrate other holidays?

Pope Pius IX has condemned such activity if it is religious in nature.

Pope Pius IX, Graves ac diuturnae (# 4), March 23, 1875:

“They [the faithful] should totally shun their religious celebrations, their buildings, and their chairs of pestilence which they have with impunity established to transmit the sacred teachings. They should shun their writings and all contact with them. They should not have any dealings or meetings with usurping priests and apostates from the faith who dare to exercise the duties of an ecclesiastical minister without possessing a legitimate mission or any jurisdiction.”

God is truth and those who worship Him must worship in spirit and truth. John 4:24. Celebrating anything with roots outside of what God deemed as acceptable (Christmas, Easter, etc., as both have pagan roots) is detestable (Leviticus 18:30, Dueteronomy 7:25,26; Dueteronomy 18:9) in the sight of God. Yes, Christ came to fulfill the law so we are no longer judged by it, but why would a christian want to commit an act that is detestable to God? I challenge all christians to identify the real reasons we want to celebrate secular holidays and use them to "thank or glorify God." If we are honest with ourselves we will realize it is not out of obediance to God, but a desire to fit with the secular world and our family, friends and memories attached to these holidays. I urge all to read what happened to the sons of Aaron (Nadab and Abihu) when as chosen priests they offered to God as worship what God did not deem as acceptable in Leviticus 10:1,2. God is the same yesterday, today and forever (Hebrews 13:8) and the truth is... God accepts worship how He wants because He can. For us to jusify attitudes, beliefs, and actions that are not scriptural is offensive to the one who created us if for nothing other than the sole reason we have elevated our views above His.

The first Christians were Jewish. In fact, Christianity is really "fulfilled" Judaism, in that the fulfillment of the Messianic prophecies are found in Jesus. So, Jewish Christians (or Messianic Jews) celebrate Jewish holidays today and always have--even from the time of Christ.

There is no prohibition to non-Jewish believers regarding the celebration of other holidays, and Christianity is not about prohibitions anyway, but about forgiveness and redemption through faith in Jesus. It is true that we are called to holy living, so as long as the celebration of holidays does not impede holiness, there really isn't any problem with that.

Of course, the celebration of other deities would be prohibited whether or not it is done on a holiday or not.

This could be considered an area where it depends on individual conscience, like eating meat sacrificed to idols that Paul mentions.

As a footnote, Christmas and Easter can be celebrated in completely secular ways, which I would say is not wrong, but definitely not Christian. If Christmas is focused on getting gifts and a fat man in a red suit, there's really nothing Christian about that. And rabbits and eggs certainly do not qualify for a celebration of the single most important event in all Christendom. So, even celebrating religious holidays does not mean that those are sacred observances, if they are done in a secular manner.

Nature worship is considered grave matter (i.e. sinful) because of the first commandment no gods before Me.

Barring that traditions, patriotism and culture are compatible with Christian living. Christianity grew up out of tradition and takes its form in the various parts of the earth because of the local traditions.

That being said, follow your conscience, intemperate celebration is a good way to forget about God. If you are required to attend Mass or your services on a holiday, put God first and attend. We've always worked to Christianize pagan holidays in the past, there's no reason we can't continue to do it in the 'multi-cultural' world.