I Swear To Tell The Truth, So Help Me God

Perhaps they will swear on the Quran

They can still swear on the Bible, and/or Quran. It is the oath before God that is important, not the related book which is more custom than obligating object. Muslims consider it "'another' Holy Book" and largely about the same God in whom they believe and trust.  It is a very complex issue, but abstracted:  Jews, Muslims, and Christian all worship the same Father/Creator God; and with Christians recognizing His other two manifestations..."Flesh" as "Son" and "Spirit" as Teacher and Comforter...all three always existing as part of a God for whom our linear concept of time, dimensional concept of space, and simplistic concepts of relationship do not have any impact.  Where they most greatly differ is in the prophet(s) they believe the most and where they put the emphasis on linear heritage, maternal for Jews and Muslims with paternal milestones such as the Jewish King David, Christian Messiah Jesus (considered incarnation of God the Father and hence "The Son") through a human (physical) mother, Mary,  and the Islamic prophet Mohammed, if I understand correctly, which I may not.  The Jews and followers of Islam also differ on who the inheriting son of Abraham was based on wife standing and maternal relationship and sequence of birth...Jews saying Isaac because Sarai was wife #1  and Muslims saying Ishmael because he was born first, despite being born to wife #2, Hagar, who was also loved by Abraham.  Abraham, did, however, acknowledge the "priviledge" and higher wifely "status" of Sarai when he had Ishmael's mother Hagar sent into the desert with her son Ishmael and kept Isaac with him in his camp as his chosen son who was the result of God's promise rather than Ishmael who was the result of Sarai and Abraham lacking faith in God's ability to keep his promise of an inheriting son through her in her old age...God's miraculous confirmation, not unlike Jesus, who Isaac foreshadowed and God promised and testified to through the prophets. Our (humans) impatience with God and trying to do things with our own known resources, in our own ways, and in our own timing,  always results in problems. When will we learn to just seek God and wait on His will to be manifested.  I think all three of these faiths need to incorporate the concept (but not the logic or religiosity) of patience embodied in Buddhism. 

It is a sibling rivaly that is thousands of years old and will not be settled or come to peace outside of heaven and God's judgement. 

I agree with the first few sentences of Leedtex. I agree with Njoy that the court will require the use of the personally preferred holy book of the witness or defendant. After all, a non-theist would be allowed to choose no book. 

I belive Njoy and Leedtex are right, the courts will most likely have them swear on their own holy book, istead of the bible.

On the subject of  whether it would be accepted or not, is a sticky one. Church and state are suppose to be separated in the country, so a Quran should be accepted by the American court system. But at the same time it is a highly Cristian nation that we live in, so the question changes into whether or not swearing by the Quran can be believed by the mostly Cristian judge and jury. In my view if these people where insane enough to attack America because their twisted views of the book where so strong, then  if they swear by the same book, then it they should respect that swear. But then one must remember they are terrorist , So you can't trust much of anything they say.

To my way of thinking(which I understand can be wrong) Seperation of church and state simply means that the US govt cannot make one religion the dominate and only religion.  No State Religion.  The kind they had in England at the time our forefathers wrote the Declaration of Independence.

Which was The Church of England invented by Henry VIII in defiance of the Catholic Church. 

While many courts will customize oaths to acknowledge non-Christian beliefs, it's simply easier to administer a secular "affirmation" that the testimony will be truthful, under penalty of perjury.

Be thankful that these are modern times.  According to some sources, men used to place their hand on their testicles instead of the Bible when vowing to tell the truth.  (Testify = testicle)