All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
First of all our answer to the questioner who is questioning why the Jews and Christians are described as non-Muslims and disbelievers and that their religion is false, is that Islam requires that one submits himself to the Law of Allaah and His Prophet . Allaah in the Qur'an and His Prophet in his Sunnah ruled that the Jews and Christians are non-Muslims and disbelievers and that their religion is falsified and distorted. What remained from these religions has been abrogated by the advent of Islam which was revealed to Prophet, Muhammad .
Allaah says (which means): {And whoever seeks a religion other than Islam, it will never be accepted of him, and in the Hereafter he will be one of the losers.}[Qur'an 3:85]. Allaah further says (which means): {Surely, in disbelief are they who say that Allaah is the Messiah, son of Maryam (Mary)….}[Qur'an 5:17]. In another verse Allaah says (which means): {Surely, disbelievers are those who said: "Allaah is the third of the three (in a Trinity)."….}[Qur'an 5:73]. Another verse reads: {Verily, those who disbelieve (in the religion of Islam, the Qur'an and Prophet Muhammad ()) from among the people of the Scripture (Jews and Christians) and Al-Mushrikuun (polytheists, idolaters, disbelievers …) will abide in the Fire of Hell. They are the worst of creatures.}[Qur'an 98:6].
Moreover, the Prophet said: "I swear by Allaah in whose hand my soul is, no one hears about me be him a Christian or a Jew, then dies not believing in what I was sent with, except that he will be among the people of Hell." [Muslim]
Furthermore, Allaah says (which means): {And We have sent down to you (O Muhammad ) the Book (this Qur'an) in truth, confirming the Scripture that came before it and Mohayminan (trustworthy in highness and a witness) over it (old Scriptures). So judge between them by what Allaah has revealed...}[Qur'an 5:48]. So all the above texts are a clear evidence that the Jews and Christians are described as non-Muslims and disbelievers and that whoever follows a religion other than Islam will be one of the losers. Whoever does not believe in the Prophet is in fact denying all the Prophets who came before him, as they were given glad tidings about him in the previous books, the Torah and the Gospel. Disbelieving in the Prophet is disbelieving and denying the books that gave glad tidings about his coming.
Giving the evidence of the verse mentioned in the question that the Jews and Christians are submitting themselves to Allaah is not correct. Indeed the verses of the Qur'an are not interpreted by desires or ignorance, but are interpreted by other verses of the Qur'an, by the sayings of the Prophet and the interpretation of the companions and according to the meaning of the Arabic language.
The Qur'an has mentioned that the people of the book are disbelievers [and non-Muslims] as mentioned in the previous verses, because if they are submitting themselves to Allaah as the questioner claims, they would believe in the Oneness of Allaah and that Muhammad is His Messenger as he was ordered by Allaah to say (which means): {Say, [O Muhammad ], "O mankind, indeed I am the Messenger of Allaah to you all, [from Him]}[Qur'an 7:162]. Indeed Allaah ordered them to believe in the Prophet as Allaah says (which means): {So believe in Allaah and His Messenger and the light [i.e. the Qur’an] which We have sent down.}[Qur'an 64:8].
One should not doubt the disbelief of the Jews and Christians, because doubting this is in itself disbelief and takes the person out of the fold of Islam. Because by thinking this way one is denying Allaah Who ruled that they are disbelievers, and denying His Prophet and the consensus of the Muslims. So whoever doubts the fact that they are disbelievers is a disbeliever, even if he prays, fasts and claims to be a Muslim.
As regards saying that they did not grow in a Muslim environment, then this is not an excuse, because they have heard about the Prophet and that he was sent as a Prophet [was given the Message] and heard about the religion of Islam which is one of the fastest growing religions in the world. So they are obliged to seek the truth and not be preoccupied by their desires and then say that they are excused.
Concerning those who did not hear about the call to Islam at all, then Allaah will not wrong them; refer to Fatwa: 85624 regarding the ruling of those whom the call to Islam has not reached.
Finally, it should be mentioned that the Muslims should carry out the duty and the responsibility of calling people to Allaah as He ordered them to do so as He says (which means): {Let there arise out of you a group of people inviting to all that is good (Islam), enjoining Al-Ma'rûf (i.e. Islamic Monotheism and all that Islam orders one to do) and forbidding Al-Munkar (polytheism and disbelief and all that Islam has forbidden). And it is they who are the successful.}[Qur'an 3:104], and He made them witnesses over them.
Allaah knows best.