The main problem is that it contradicts the Qur'an itself. In various places Muhammad admitted he could not or did not have to perform miracles to prove himself. When asked to perform miracles his response was:
Glory be to my Lord; am I aught but a mortal messenger?
Qur'an 17:93
In another place Muhammad put the following words into the mouth of Allah:
And those who disbelieve say: Why has not a sign been sent down upon him from his Lord? You are only a warner.
Qur'an 13:7
Muhammad insisted that even though other prophets had miracles, his only miracle is the Qur'an. So when Muslims claim that Muhammad performed miracles, they are going against the Qur'an itself. (Please see: Muhammad's Miracles?)
The other problem with this claim is that such a phenomenon should have been observed not just in Mecca but also all over the world. Many cultures have recorded even eclipses, why then are there no mentions of this, which surely would have been the most amazing phenomenon?
Moiz Amjad, a scholar that answers questions for Understanding-Islam.com says (here), “I have not yet come across any historical accounts of other nations, which refer to the incident.” He thinks this is “a sign of the promised hour.”
Curiously the same site contradicts itself (here) and claims that in fact a certain king of India observed the splitting of the Moon and he sent his son to Mecca who converted to Islam and upon returning he died in Yemen.
This story is of course another fabrication of Muslims (as proven here). Muslims are fond of fabricating stories to attribute Miracles to their prophet. They do it all the time. One such ridiculous fabrication was the claim that the astronauts who landed on the Moon heard the adhan (Muslim call to prayer) being chanted there. (Please see: Fake Conversions - Neil Armstrong) There are thousands more.
First of all there is no record that any King of India ever visited Muhammad. This is not mentioned in any biography of Muhammad.
Secondly the splitting of the Moon should have been observed by everyone all over the world and not just by one king. Where is the record of such a phenomenon?
Thirdly, assuming that this Indian king saw the splitting of the Moon, how could he interpret this alleged cosmic phenomenon as the sign that there has appeared a new prophet in Mecca? Muslims claim that Indian scriptures contain predictions about the coming of a messenger from Arabia. This is bull. There is no mention of Muhammad in any sacred book of any religion.
Many Muslims are convinced that this phenomenon has indeed happened. They are shown these close-up pictures of the Moon taken by NASA as the evidence and as gullible as they naturally are, they believe without any hesitation.

