Christians most often make comments stating that Muslims worship the black stone in the Kaaba and that is why they kiss it. This is absolutely false. In order to get a better understanding I will give you some background and context.
The scholars of Islam have differed on who first built the Kaaba. Some say that it was built by the angels. Others say the father of humankind, Adam first built the Kaaba but over many centuries it fell into disrepair and was lost in the mists of time, to be rebuilt by Prophet Abraham and his son Ishmael on the command of God. All agree that the Kaaba was either built or rebuilt by Prophet Abraham (pbuh).
“And (remember) when Abraham and (his son) Ishmael were raising the foundations of the House (the Kaba at Mecca), (saying), ‘Our Lord! Accept (this service) from us. Verily, You are the All-Hearer, the All-Knower’” (Quran 2:127)
God commanded Abraham to do the tawaf. The act of worship which is particularly associated with the Kaaba, and never stops except when congregational prayer is held, is tawaf, which means walking round the Kaaba seven times in an anti-clockwise direction. Tawaf is one of the duties of Islamic pilgrimage and Umrah (mini-pilgrimage). He was not able to keep a correct count of the rounds he made. He felt that other worshippers would be similarly confused. He prayed God to give him a sign to be used for counting rounds. The Angel Gabriel brought him the black stone to be used as the point of the start of the rounds.
The Prophet (peace and blessings of Allah be upon him) kissed the Black Stone, and his ummah followed his lead in doing so.
It was narrated that ‘Umar (may Allaah be pleased with him) came to the Black Stone and kissed it, then he said: “I know that you are only a stone which can neither bring benefit nor cause harm. Were it not that I had seen the Prophet (peace and blessings of Allaah be upon him) kiss you, I would not have kissed you.”
(Narrated by al-Bukhaari, 1520; Muslim, 1720)
So the stone basically shows Muslims the start of each circumvention around the kaaba and helps keep count in the completing the 7 tawaf’s. Muslims will either touch or kiss the stone on each round if they can reach it or simply point to it saying ‘In the Name of God. God is Great. (Bismillah Allahu Akbar).
The reason that the Prophet (pbuh) kissed the black stone was only because it was brought from heaven by the angel Gabriel in respect, veneration and nothing else.