In order to answer this question precisely, a few points need to be pointed out:

1- The splitting of the moon is one of the great miracles of Prophet Muhammad (pbuh). In surah Inshiqaq, Allah (swt) speaks of this magnanimous incident in such a manner: “اقْتَرَبَتِ السَّاعَةُ وَ انْشَقَّ الْقَمَرُ”[1]. In this verse, two important incidents have been mentioned; one being the nearness of the Day of Judgment which is the greatest change the universe is to undergo, and the other being the significant miracle of splitting the moon which is both a manifestation of Allah’s (swt) never-ending power and proof of Prophet Muhammad’s (pbuh) prophethood.[2]

There is more than enough evidence that proves this was a miracle, namely: This matter has been mentioned in the Quran using a past tense verb, showing that the splitting of the moon has taken place, the same way the Day of Judgment has neared as a result of the coming of the final prophet (pbuh).

Add to that the many hadiths in Islamic references that narrate this incident as a miracle of the prophet (pbuh). These hadiths are so high in number to the extent that they leave no doubt that the splitting of the moon was truly a miracle of the prophet (pbuh).

What is more interesting is the reason why this miracle took place. The disbelievers said that magic and sorcery only have an effect on the earth, so in order to be sure of the prophet (pbuh) being truthful in his claim and that his miracles aren't plain magic, they proposed that he split the moon.[3]

2- The splitting of the moon and contemporary science

First off, not only isn't the splitting and breaking of celestial bodies possible according to astrology and science, but takes place a lot too, although the causes of such phenomena may vary.

To put it another way, splitting, breakage and explosions frequently occur in space; for instance, scientists today say that the solar system came about through an explosion that involved the sun, breaking parts of it off, hence the different members of the “family” of the solar system.[4]

Anyway, the splitting of celestial bodies isn't something that hasn’t happened before, and isn't considered scientifically impossible, so it can't be said that something impossible was the subject of a miracle.

As for the moon coming together again after splitting, it is because of the gravitational force between the two halves, although this fact isn't correct according to the Ptolemaic system of astrology, because according to this system, our solar system is made up of nine spheres that are stuck together like the layers of an onion, making their splitting and coming together impossible, and it is because of this that the advocates of this system didn’t believe in bodily ascension to the skies and the splitting of the moon, because it necessitated the piercing of these spheres. But today, this system isn't followed anymore and is considered an incorrect and out-of-date one so its claims on the splitting of the moon being impossible are disregarded.[5] According to NASA, our moon has indeed split before and come together again. The official website of Al-Watan, an Arabic newspaper printed in the US, has quoted a Jordanian astronomer and space specialist saying: “After years of orbiting the moon and doing research, the space probe “Clementine” has reached the conclusion that the moon has split into two halves hundreds of years ago, and come together again.” This astronomer has sent a report to NASA explaining that Muslims believe this phenomenon belongs to 1400 years ago and is a miracle of their prophet; a phenomenon that NASA still hasn’t been able to discover the true cause to, because it is a unique one that hasn’t taken place for other celestial bodies.[6]

Two more points to consider:

1- Proving Islamic facts that are somewhat metaphysical, isn't possible through normal science that is limited to this material world and is only physical, especially since it has developed in an environment that has had nothing to do with Islam.

2- If an issue is logically possible, without any reason for its impossibility, and is something that Islam firmly says has taken place, there is no longer any need to prove it scientifically; although it is sometimes better to do so, so that one becomes personally sure of the fact’s truth.

Here are two sites that have discussed the issue of the splitting of the moon:

1- http://www.nojumi.org/newsn/news.php?action=fullnews&id=1095

2-http://www.cloob.com/club/article/show/clubname/croobank/articleid/111661/redirectkey/1c8c2157f460636134e7fd10eac9b37c