Answer
Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
The Prophet Muhammad spread Islam through these methods:
1- Giving an example from himself (peace be upon him).
2- Educating his companions to be role models for humanity.
3- Teaching the concept of dawah (call for Islam).
Answering your question, Dr. Jasser Auda, Professor and Al-Shatibi Chair of Maqasid Studies at the International Peace College South Africa, states:
God had sent the Prophet (peace be upon him) as a “mercy for all the worlds” (Al-Anbiya’ 107). The methods that he (peace be upon him) followed in order to spread this message to humanity could be summarized into the following:
Giving an example from himself (peace be upon him);
Educating his companions to be role models for humanity;
Teaching the concept of dawah (call for Islam).
Regarding the first method, the Prophet (peace be upon him) spread the message of Islam through showing humanity an example of an excellent human being.
He was an ideal teacher, friend, husband, father, grandfather, worrier, speaker, neighbour, employer, etc.
In all the roles that he played in his life (peace be upon him), he gave humanity examples in every aspect and in every sense. And this is the main means through which the message of Islam was spread amongst his people.
When Jafar Ibn Abi Talib (may God be pleased with him) was asked by Al-Najashi, the King of Ethiopia at the time, about the message of Muhammad, Jafar said: ‘He (Muhammad) is a man who is asking us to be honest, chaste, and good to our relatives.’ When Aisha, the Prophet’s wife, was asked about his manners, she said: ‘His manners were the Quran.’ (Al-Bayhaqi)
God Almighty described His messenger (peace be upon him) by saying: {For, behold, you keep indeed to a sublime way of life.} (At-Taghabun 68:4)
He (peace be upon him) told his companions that a good character is the best way to win people’s hearts. He said: “You will never make people pleased with your money, but you will win their hearts with a smiley face and good manners.” (Al-Hakim)
The second factor in the spread of Islam was the way the companions were educated by their teacher (peace be upon him), in order to be role models themselves.
Islam was spread in various parts of the world through the companions excellent conduct, which they spread through their travels after the Prophet (peace be upon him) passed away.
The Islamic state did not force people to become Muslim by any means (except for some mistakes that had happened by a few ignorant rulers).
The way Islam spread in these vast areas in Asia and Africa was basically through the excellent example that the companions (and their students) gave to people.
The areas of the largest Islamic populations today (in East Asia) had never seen any military conflict with any Islamic state.
Indonesia is a clear example. The way Islam spread and, later, became the religion of the overwhelming majority of that country was through good manners.
The third method through which the Prophet (peace be upon him) spread the message of Islam was by teaching the concept of dawah (call for Islam).
The Prophet (peace be upon him) taught his companions, and all Muslims, that, ‘A caller for a good deed is rewarded a reward equal to the reward of those who follow him in his/her call.’ (Muslim) He also said: ‘convey the message after me, even one verse.’ God describes the Prophet’s way by saying:
{Say [O Prophet]: “This is my way: Resting upon conscious insight accessible to reason, I am calling [you all] unto God – I and they who follow me.” And [say:] “Limitless is God in His glory; and I am not one of those who ascribe divinity to aught beside Him!} (Yusuf 12:108)
So, these are the three methods through which the Prophet of Islam spread the message of Islam.
And by the way, regarding the part of your question where you mentioned that Prophet Muhammad (peace be upon him) had ‘translated the message and wrote it in the Quran,’ I would like to note that there is a bit of inaccuracy in this expression.
Prophet Muhammad (peace be upon him) received the message of the Quran as we know it today, in the same (Arabic) language. Some people suggested that he (peace be upon him) ‘wrote the Quran’ himself, based on some ‘coded message’ that he received from the Angle.
However, many sayings of the Prophet and verses of the Quran are clearly stating that he (peace be upon him) received the verses as they are, that is, in the same Arabic script that we know today. God said, for a few examples:
{Behold, We have bestowed it from on high as a discourse in the Arabic tongue, so that you might encompass it with your reason.} (Yuauf 12:2)
{And thus have We bestowed from on high this [divine writ] as a discourse in the Arabic tongue.} (Taha 20:113)
{…a divine writ, the messages whereof have been clearly spelled out as a discourse in the Arabic tongue.} (Fussilat 41:3)
{[You are but entrusted with Our message:] and so We have revealed unto you a discourse in the Arabic tongue, in order that you may warn the foremost of all cities and all who dwell around it.} (Ash-Shura 42:7)
{… and this [Quran] is a divine writ confirming the truth [of the Torah] in the Arabic tongue.} (Al-Ahqaf 46:12)
Almighty Allah knows best.