"In revenge for an ambush on some Muslim missionaries, Muhammad sent Amr bin Umayya and a companion to assassinate Abu Sufyan, a leader of the Meccans. This shows that the prophet could get caught up in the cycle of violence that went on endlessly in seventh-century Arab culture. Umayyah failed in his attempt, and he had to flee under pursuit, hiding in a cave, murdering a man named Ibn Malik along the way. As the pursuit was dying down, a tall, one-eyed, unnamed Bedouin entered the cave, driving some sheep. Umayyah and the Bedouin introduced each other. After they settled down, the shepherd sang a simple two-line song in defiance of Muslims and Islam:
I will not be a Muslim as long as I live,
And will not believe in the faith of the Muslims. (Watt)
Another translation reads:
I won’t be a Muslim as long as I live,
Nor heed to their religion give. (Guillaume)
Unfortunately for this Bedouin, he was in the cave with a radical Muslim, who said: "You will soon see!" The Bedouin fell asleep, snoring. Umayyah recounts what he did: ... "I went to him and killed him in the most dreadful way that anybody has ever been killed. I leaned over him, stuck the end of my bow into his good eye, and thrust it down until it came out of the back of his neck." He fled back to Muhammad, who said, "Well done!" The account ends: The prophet "prayed for me [Umayyah] to be blessed."
When I looked in my books, for a similar account like the one given, I couldn't find it. This made me think about the authenticity of the above, so I doubted if the story was true or not. Tabari relied heavily on the works of Ibn Ishaq, however, Al-Dhahabi said that Ibn Ishaq used to borrow reports from Christians, and Jews:
" Al-Dhahabi also declares that Muhammad ibn Ishaq reported facts borrowed from the Jews and the Christians, whom unforutnately, he considered as reliable."
(Source: Sirat Un Nabi by Allama Shibli Nu'Mani rendered into English by M. Tayyib Bakhsh Budayuni Vol.I, p.22, Kazi Publications Lahore Pakistan)
"But let alone this logical criticism, we shall have to admit that the story even if judged on the basis of reports, is unacceptable. Sahih-Al-Bukhari mentions the execution of Ibn Khatal alone, and this is admitted on all hands that he was executed for a murder. The execution of Miqyas too was a retaliatory sentence. All such reports, as ascribe the execution of others merely to their having harrased the Prophet (peace and blessings of Allah be upon him) in the past, have Ibn Ishaq as the last narrator at the top; and in the terminology of the traditionalists such reports are called Mursal and are not to be relied on."
(Source: Sirat-Un-Nabi, by Allama Shibli Nu'Mani, rendered into English by M. Tayyib Bakhsh Budayuni, Kazi Publications Lahore, Vol. II, p. 199-203, bold and underlined emphasis ours)
Therefore it is clear that this story is not from a aunthentic source on Islam and as such it is not true. Sadly however the anti-islamists continue to ignore this and propogate it for their own cause. You will never find support for such action in the Quran, and such you will never see an anti-islamist refer to the authentic sources of Islam to make their case.