Is there a Hadith numbered 6324 in any of the Hadith books, which speaks of obscene (vulgar) words uttered by the Prophet, peace and blessings be upon him, and of the Prophet's command to stone a man who came to him repentant of adultery, knowing that Christians use this Hadith to cast doubt on the religion?

The hadith you cited, numbered in Sahih al-Bukhari and other collections, narrates the story of Ma'iz ibn Malik and his stoning after he confessed to adultery. It indicates the Prophet’s — peace and blessings be upon him — compassion and his attempt to prompt Ma'iz with an excuse to escape stoning. The Prophet — peace and blessings be upon him — commanded concealment of sins, for concealment is better than exposure. Furthermore, the spread of indecencies leads to the outbreak of epidemics, which modern science has confirmed with the spread of diseases like AIDS due to adultery. Defending adulterers is an act of obstinacy, for adultery harms the individual and society. The wise Sharia differentiated in punishment between the married adulterer (muhsan) and the unmarried adulterer. One should not be swayed by the doubts cast by Christians about our religion, for they will never be satisfied until you follow their creed.