Islam emerged in Asia and it spread across the Arab world and Asia. There are multiple factors behind the fact that Islam did not spread in Europe. Those factors include the internal discord and conflicts among Muslims themselves and their failure to introduce the true and beautiful image of Islam to people in Europe.
Following the demise of the Prophet of Islam (s), efforts were made to preach and spread the religion of Islam but Muslims did not make any progress due to the fact they were engulfed in internal conflicts. However, efforts were renewed and Muslims began from the year 71 to 398 of the hegira year to spread Islam. During this time, Africa and some parts of Europe had been conquered by Muslims. Islam spread in these two continents to some extent. But in our time Islam is developing and spreading rapidly in Europe and other countries.
The topic concerning the development of Islam requires a lengthy answer and detailed discussion. However, we shall suffice to giving a brief answer to the above question as below: Islam emerged in Asia (Mecca in the Arabian Peninsula) during the 7th century A.D. It commenced with the revelation sent down to the Prophet Muhammad. Since Islam emerged in Asia, it spread across Arabic countries and Asia. There are multiple factors behind the fact that Islam did not spread in Europe. Those factors include the internal discord and conflicts among Muslims themselves and their failure to introduce the true and beautiful image of Islam to people in Europe.
To further explain, following the demise of the Holy Prophet (s) the Muslim army moved out towards the east and west of the Arabian Peninsula. The contingent that moved eastward was headed by Sa'd Waqas and the one that moved towards Syria and Palestine was headed by a group of renowned figures like Umar bin Al'As, Yazid bin Abi Sufyan, Abu Ubaidah bin Al-Jarrah, Sharhbil bin Hasanah and Khalid bin Walid. The Muslim army reached Persia but the army that moved westward did not make headway because Muawiyah bin Abi Sufyan had signed a treaty with the Roman Empire that was meant to avoid launching any attack on either side. The Muslim could not advance westward because of the same treaty.
In addition, the transition and conversion of the caliphate to monarchy by Muawiyah caused the West not to understand the true nature of Islam. Thus, the people had no desire to welcome and embrace Islam there in Europe. One of the prominent German scholars says: "We should make a golden statue of Muawiyab bin Abi Sufyan and put it up in Berlin, the German capital because if it he did not inflict a mortal blow on Islam, Islam would have taken all over the world and we, the Germans, and people in other European countries would have been Arabs and Muslims.
But during the government of Walid bin Abdul Malik, Muslims from Iraq, Iran, Syria, Egypt and Africa crossed the Mediterranean Sea, reached Tariq Mount (Jabal al-Tariq) and conquered Europe as far as the Atlantic shores. In the year 711 A.D. Muslim conquered Spain and the message of Islam reached there and spread out also to other European countries.
Also, Muslims were able to cross Seyhoun and Oxus and were continuously delivering Islam to the people of the cities and countries. Eventually, the great Islamic empire came into being with its capital in Damascus, Syria.
Therefore, in the year 78 A.H. the entire Europe and Africa had been conquered by Muslims. Islam dominated Europe until 398 A.H. when Abdul Malik Mansoor died and his brother Abdu Rahman titled Al-Naser Le Deenillah came to power. He also, like his father and brother, ruled Europe without any belief in the caliph of his time, Hesham bin Hakam. He decided to eliminate the remaining caliphate norms and customs. For this reason, he asked Hesham to appoint him as his crown prince.
Many people were unhappy with his becoming a crown prince. That was why family and internal disputes began among Muslims engaging them and keeping them unmindful of the borders. Feudalism came into being within the Islamic state causing Muslims to fight one another while their enemies were watching them. The disputes among Muslims and the wrong policies adopted by Muslim rulers gave Christians the opportunity to expel them from Europe in the wake of several wars.
However, in recent years especially in the present time, the Islamic culture has spread all over the world including Europe. We should not neglect the role of European countries' propaganda machine and their highly advanced facilities in presenting a distorted image of Islam and in preventing the voice of Islam to reach people. The fact that they are trying to prevent Islamic influence is an independent topic that needs to be discussed in its own appropriate time and occasion.
Another probability that is necessary to mention is that the Romans believed in Christianity in that time. Obviously, according to them in that time, Christianity was considered to be a divine religion. With such a belief and conviction among the Romans, it was very difficult for Muslims to go into a war with them.
If Muslims were to fight Christians, they had to fight them on two fronts: One, the scientific and cultural front which would require Muslims to explain the righteousness of Islam and expose the deviation of the Christian religion from the right path, second, the military front to defeat the enemy militarily. Obviously, the Amawid government had not trained such people to defend their religion in these areas.
In addition, the teachings of the school of the Ahlulbayt, pace be upon them, did not advance to those regions due to the fact that the Amawid government was strongly opposed to them. Unlike Europe, Islam rapidly progressed in Iran and other countries in Asia because people got acquainted with this advanced school. Thus, not only did people embrace it but they also defended and disseminated its teachings here and there. Perhaps, the Abbasids' motto (Reza is from the family of the Prophet) is the best evidence proving out point.
 - Kufi, Ahmad bin Ali bin A'tham, Al-Fotuh, pg. 182 – 305, translated by Mustawfi Herawi, Muhammad bin Ahmad, Scientific and Cultural Publications 1374 (1995).
 - Salehi Najaf Abadi, Shahid Jawid, pg. 313, 8th edition.
 - Shakib Arsalan, Tarikh Fotuhat Islami dar Uropa (History of Islamic Conquests in Europe), Dawani, Ali, pg. 37, Bani Hashem Bookstore.
 - Ibn Hesham was a person other than the famous Hesham bin al-Hakam, the Amawid ruler. In fact, he had been "al-Moayyid billah", one of the rulers of Andalusia. Al-E'lam, vol.2, pg. 310.
 - Ibid, pg. 295.