From the scholars I have read, the narrations which depict this event as: the Prophet calling the Quraish to come to him, then he raising his index finger, due to which the Moon gets split into two, with such a distance between the two portions that one of its parts gets hidden behind a mountain - are unreliable narrations. The companions to whom such narrations are attributed were either not present or unable to be witness to the event. The most significant personality among them is Ibn Abbas, but the huge and obvious problem with Ibn Abbas being a witness is that, from other sources we know, he was around 13 years old when the Prophet died and this event happened during the Makkan period which means that during this event Ibn Abbas was 1-3 years old therefore making him unable to be a witness to this. Apparently, because of the high status of Ibn Abbas, he has often become the preferable "narrator" of fabricated narrations.
Some scholars, even though they believe in the splitting of the moon, contend that this wasn't a miracle of the Prophet because he didn't call people to come to him and told them he is going to show them a miracle but it was an event (actual splitting of the moon or whatever) that happened during the Prophet's time in Makkah and he directed the attention of the Quraish towards it.
No, the hadiths are authentic. I don't know why everybody is against the moon split so much.
Narrated Ibn Masud:
During the lifetime of Allah's Messenger (ﷺ) the moon was split into two parts; one part remained over the mountain, and the other part went beyond the mountain. On that, Allah's Messenger (ﷺ) said, "Witness this miracle."
Grade: Agreed Upon
Source:
USC-MSA web (English) reference : Vol. 6, Book 60, Hadith 387
Arabic reference : Book 65, Hadith 4864
There is atleast a bit of proof of Moses PBUH splitting the sea,as mentioned in the Qur'an AND HADITH,
but there is no Proof for Muhammad PBUH splitting the moon.
There is proof from both the Quran and the hadith of the moon splitting. "The Hour has come near, and the moon has split [in two]." 54:1
and in the Muhsin Khan translation:
"The Hour has drawn near, and the moon has been cleft asunder (the people of Makkah requested Prophet Muhammad SAW to show them a miracle, so he showed them the splitting of the moon)." 54:1
and the authentic hadith above.
You should also read the tafsirs:
Tafsir Ibn Kathir Surat Al-Qamar verse 1:
وقوله : ( وانشق القمر ) : قد كان هذا في زمان رسول الله - صلى الله عليه وسلم
And his saying ( the moon has split [in two].): This was at the time of the prophet (peace be upon him).
Source:
http://quran.ksu.edu.sa/tafseer/katheer/sura54-aya1.html