There is a black stone in the corner of the Kaaba, which Muslims circumambulate, point to, touch, and kiss. These practices regarding the Black Stone are taken directly from the example of Muhammad, as we read:
Sahih al-Bukhari 1603
Narrated Salim that his father said:
I saw Allah's Messenger (ﷺ) arriving at Mecca; he kissed the Black Stone Corner first while doing Tawaf, and did ramal in the first three rounds of the seven rounds (of Tawaf).
Other narrations of Muhammad kissing and circling (Tawaf) the Black Stone include Sahih Muslim 1218d, 1227, 1261c, 1780a, 2473a, and Sunan abu-Dawud 1879.
One of Muhammad's companions, Umar, who would go on to be the second Caliph, remarked the following about the practice:
Sahih al-Bukhari 1597
Narrated `Abis bin Rabi`a:
Umar came near the Black Stone and kissed it and said "No doubt, I know that you are a stone and can neither benefit anyone nor harm anyone. Had I not seen Allah's Messenger (ﷺ) kissing you I would not have kissed you."
This narration or similar is also given in Sahih al-Bukhari 1605, 1610, Sahih Muslim 1270a-d, Sunan abu-Dawud 1873, Sunan ibn-Majah 2943, Sunan an-Nasai 2946, and Jami at-Tirmidhi 861.
Above, Umar links the practice directly to Muhammad, and says that he would not be kissing the stone if he had not seen Muhammad do it. He also foolishly says that the stone "can neither benefit anyone nor harm anyone", which is false, because worship of stones is idolatry, which will send someone to Hell, and that is exactly what the false prophet Muhammad instituted here in Islam.
We also know from Islam's own sources that stone worship was common amongst the Pagans in Arabia during Muhammad's time. One of Muhammad's companions describes the practice:
Sahih al-Bukhari 4376
Narrated Abu Raja Al-Utaridi:
We used to worship stones, and when we found a better stone than the first one, we would throw the first one and take the latter, but if we could not get a stone then we would collect some earth (i.e. soil) and then bring a sheep and milk that sheep over it, and perform the Tawaf around it.
Above, we see that both worshipping stones, and the "Tawaf" - circling an object, as Muslims do with the black stone seven times during the Hajj - have their origin in local Arabian Paganism. The same is true of the circling of the Kaaba, and the running between the hills of Safa and Marwa.
In addition, Muhammad told people that the Black Stone came down from Heaven (Sunan an-Nasai 2935), which is as valid excuse as any for idolatry, I suppose. Of course, every stone on earth was created by God, and thousands of meteorites fall to earth every single year. There is nothing special about any particular one of them, unless you happen to be a stone-worshipping Arabian Pagan.
When it was sent down, the Black Stone was originally white, but turned black from the sins of mankind (Jami at-Tirmidhi 877). Is this stone taking our sins, Muslims? Is the stone our savior?
Also be aware that, for some reason, this stone will testify on the day of judgement as to who touched it:
Jami at-Tirmidhi 961
Ibn Abbas narrated that:
The Messenger of Allah said about the (Black) Stone: "By Allah! Allah will raise it on the Day of Resurrection with two eyes by which it sees and a tongue that it speaks with, testifying to whoever touched it in truth."
God is apparently interested in who touched this stone, which He sent down for people to touch, so that they can touch a stone. Interesting. There are 1,000 different ways Paganism is justified in the minds of its adherents, and if you desire strongly enough, even worshipping a rock can be justified.
Muhammad claims ad-nauseam in the Quran to be confirming the Old Testament (Surah 3:3, 5:68, etc.), yet this practice violates a clear command from God:
Leviticus 26:1
1 Ye shall make you no idols nor graven image, neither rear you up a standing image, neither shall ye set up any image of stone in your land, to bow down unto it: for I am the Lord your God.
It's strange how God would specifically command us not to set up a stone to worship, and then send down a stone from Heaven for mankind to worship.
Despite this being a purely and obviously Pagan, superstitious, idolatrous practice, it remains in Islam, because Islam is an amalgamation of whatever Muhammad's depraved mind pieced together when forming his new, false religion.
But, in the minds of Muslims, as long as Muhammad did it, stone worship can't possibly be idolatry, because idolatry is never idolatry when we do it, for this or that reason. Ah yes, tell us the 500 valid reasons you have for kissing and circling and pointing to a piece of rock.