Short answer
Because the message he brought is the one every people's warners brought, and he was sent to all of them. Each faith has a reason to say it needs no prophet from Arabia. Jewish tradition holds that prophecy has ended. Buddhists hold that each person must find the way alone. Daoists and Confucians hold that Heaven has already spoken to China. And many Chinese see Islam as a foreign religion. The Qur'an says God (﷾) sent a warner to every people, all with one message: worship God (﷾) alone. Then He sent Muhammad (ﷺ) to all mankind, as the last of them, with a Book that has been kept.
Full explanation
Every people had a warner
The Qur'an says there never was a people without a warner. It says God (﷾) sent a messenger to every people, with the command to serve Him alone. So Islam does not say that God (﷾) left China, India or Israel without guidance. It says the warners of every people brought one message.
“Verily We have sent thee in truth, as a bearer of glad tidings, and as a warner: and there never was a people, without a warner having lived among them (in the past).”
“For We assuredly sent amongst every People a messenger, (with the Command), "Serve Allah, and eschew Evil": of the People were some whom Allah guided, and some on whom error became inevitably (established). So travel through the earth, and see what was the end of those who denied (the Truth).”
A messenger to all people
Every earlier prophet was sent to his own people. The Qur'an says Muhammad (ﷺ) was sent to all mankind, and that he is the Seal of the Prophets, the last of them. It says God (﷾) made nations and tribes so that they may know one another. So his message does not belong to the Arabs alone, and it is foreign to no one.
“We have not sent thee but as a universal (Messenger) to men, giving them glad tidings, and warning them (against sin), but most men understand not.”
“Say: "O men! I am sent unto you all, as the Messenger of Allah, to Whom belongeth the dominion of the heavens and the earth: there is no god but He: it is He That giveth both life and death. So believe in Allah and His Messenger, the Unlettered Prophet, who believeth in Allah and His words: follow him that (so) ye may be guided."”
“Muhammad is not the father of any of your men, but (he is) the Messenger of Allah, and the Seal of the Prophets: and Allah has full knowledge of all things.”
“O mankind! We created you from a single (pair) of a male and a female, and made you into nations and tribes, that ye may know each other (not that ye may despise (each other). Verily the most honoured of you in the sight of Allah is (he who is) the most righteous of you. And Allah has full knowledge and is well acquainted (with all things).”
How to tell
Every claim to speak for Heaven must be tested. The Qur'an asks to be tested. It says that if it were from anyone other than God (﷾), people would find many contradictions in it. And it challenges anyone who doubts it to produce one chapter like it. Whether Muhammad (ﷺ) himself was truthful, mad or after gain is set out on the page "Was Muhammad (ﷺ) really a messenger of God?".
“Do they not consider the Qur'an (with care)? Had it been from other Than Allah, they would surely have found therein Much discrepancy.”
“And if ye are in doubt as to what We have revealed from time to time to Our servant, then produce a Sura like thereunto; and call your witnesses or helpers (If there are any) besides Allah, if your (doubts) are true.”
Perspectives
Jewish: prophecy is closed and the Torah does not change
Jewish tradition holds that the line of prophets ended long ago. Maimonides rules that the Torah's commandments stand for ever, and that anyone who says they were only for a time is a false prophet.
There has been no unbroken line of prophets since the days of Artaxerxes. So no later prophet can bring a new scripture.
Rebuttal
Josephus says only that the line of prophets was broken. He does not say that God (﷾) could never send another. And the Torah itself promises one: I will raise them up a Prophet from among their brethren, like unto thee, and will put my words in his mouth (Deuteronomy 18:18).
A prophet who says a commandment was only for a time is a false prophet.
Rebuttal
That is how Maimonides reads it. But the Torah's own test of a prophet is different. Does he call people to other gods? And does what he foretells come true? Muhammad (ﷺ) called only to the God (﷾) of Abraham. And the Torah tells Israel to listen to the prophet God (﷾) would raise up.
Buddhist: no messenger is needed
The Buddha found the way by his own effort. He taught that each person must do the work, and that the Buddhas only show the way. On this view, no messenger from God is needed.
You must do the work yourselves; those who have found the way only point to it. Even the verses that came to the Buddha after his awakening were, he said, neither given from above nor learned before.
Rebuttal
Then, by the Buddha's own account, his teaching was not revelation. So it cannot speak about the unseen as a messenger can. Yet the same sutta says he began to teach only after the divine being Sahampati urged him to. So Buddhism itself accepts that help can come from above. The Qur'an says God (﷾) sent a messenger to every people, and sent Muhammad (ﷺ) to all mankind.
Daoist: Heaven already spoke through Lord Lao
The Way of the Celestial Masters goes back to a revelation in 142 CE. In it, the divine Lord Lao came down to Zhang Daoling and made a new covenant. On this view, China already has its revelation.
Lord Lao came down to Zhang Daoling in 142 CE and gave him a new covenant. So China needs no prophet from Arabia.
Rebuttal
Then Daoism accepts that Heaven can speak through a chosen man. Every such claim must be tested, and one test is what it says about God (﷾). This tradition worships Lord Lao, and holds that the man Laozi was one of his forms. A true messenger calls himself only a man. And the Qur'an says of God (﷾): He begetteth not, nor is He begotten, and there is none like unto Him.
Confucian: Heaven has spoken to China already
The Book of Odes tells how the Lord on high chose the house of Zhou and spoke to King Wen. Some Confucians say Heaven has already spoken to China, so there is no need for a prophet from anywhere else.
The ode Hwang I says the Lord on high spoke to King Wen. So Heaven has already spoken to China.
Rebuttal
The Qur'an itself says every people had a warner. But the ode is a poem about the rise of the house of Zhou, not a book sent down to a messenger. Muhammad (ﷺ) was sent to all people, Chinese and Arab alike. And the Qur'an claims to be God's (﷾) word, and invites anyone to test it.
Chinese tradition: a foreign religion
For more than a thousand years Islam was seen in China as a foreign presence. Many Chinese still feel that Muhammad (ﷺ) was a prophet for Arabs, not for them.
Islam is a foreign religion, for Arabs and not for Chinese.
Rebuttal
The Qur'an says the Prophet (ﷺ) was sent to all people, and that God (﷾) made nations and tribes so that they may know each other. Islam has been in China for more than 1,200 years. The Hui are Chinese-speaking Muslims.
The Islamic perspective
God (﷾) sent a warner to every people, all with one message. Then He sent Muhammad (ﷺ), the last of the prophets, to all mankind. No people's own tradition shuts out that last message.
The Qur'an says there never was a people without a warner.
The Qur'an says Muhammad (ﷺ) was sent to all mankind, and is the Seal of the Prophets.
The Qur'an asks to be tested. It says it has no contradictions, and challenges anyone to produce a chapter like it.