According to the Qur'an the dominion of the morality of Islam over the world is an obligation

In many verses of the Qur’an Allah promises the dominion of the morality of Islam all over the world. The promise of our All-Mighty Lord is true and Allah never fails to keep His promise. Those who believe in Allah also believe that Allah’s promise will absolutely come true. Because our All-Mighty Allah gives the news of the dominion of the morality of Islam over the world and promises it, what befalls to sincere believers is to pray for its realization by heart, demand it and strive for it with all their might. Believing that morality of Islam will reign and making effort for the realization of this blessed event is obligatory for every Muslim who abides by the Qur’an. In one verse our All-Mighty Allah says:

Allah has promised those of you who believe and do right actions that He will make them successors in the land as He made those before them successors, and will firmly establish for them their religion with which He is pleased and give them, in place of their fear, security. ‘They worship Me, not associating anything with Me.’ Any who are unbelivers after that, such people are deviators. (Surat an-Nur, 55)

Some other verses in which Allah promises the dominion of the morality of Islam over the world are as follows:

Allah has written, ‘I will be victorious,    I and and My Messengers.’    Allah is Most Strong, Almighty. (Surat al-Mujadala, 21)

They desire to extinguish Allah’s Light  with their mouths but Allah will perfect His Light, though the unbelievers hate it. It is He Who sent His Messenger with guidance and the Religion of Truth to exalt it over every other religion, though the associaters hate it. (Surat as-Saff, 8-9)

They desire to extinguish Allah’s Light with their mouths. But Allah refuses to do other than perfect His Light, even though the kafirun detest it. It is He Who sent His Messenger with guidance and the Religion of Truth to exalt it over every other deen, even though the associaters detest it. (Surat a-Tawba, 32-33)

Allah confirms the Truth by His words, even though the evildoers hate it. (Surah Yunus, 82)

Those before them plotted but all plotting belongs to Allah. He knows what each self earns, and the unbelievers will soon know who has the Ultimate Abode. (Surat ar-Ra‘d, 42)

We destroyed generations before you when they did wrong. Their Messengers brought them the clear signs, but they were never going to have faith. That is how We repay evildoers. Then We appointed you after them to be successors on the earth so We might observe how you would act. (Surah Yunus, 13-14)

That those people of pure natural belief in Allah who do not associate any partners to Him will be the inheritors of this world is a divine Law which is stressed in many verses of the Qur’an:

We wrote down in the Psalms, after the Reminder came: ‘It is My servants who are righteous who will inherit the earth.’ (Surat al-Anbiya, 105)

We will leave you the land to live in after them. That is the reward of those who fear My station and fear My threat.’ They asked for Allah’s victory, and every obdurate tyrant failed. (Surah Ibrahim, 14-15)

Moses said to his people, ‘Seek help in Allah and be steadfast. The earth belongs to Allah. He bequeathes it to any of His slaves He wills. The successful outcome is for those who fear [and respect Allah].’ They said, ‘We suffered harm before you came to us and after you came to us.’ He said, ‘It may well be that your Lord is going to destroy your enemy and make you the successors in the land so that He can see how you behave.’ (Surat al-A‘raf, 128-129)

We destroyed generations before you when they did wrong. Their Messengers brought them the clear signs, but they were never going to have faith. That is how We repay evildoers. Then We appointed you after them to be khalifs on the earth so We might observe how you would act. (Surah Yunus, 13-14)

Rather We hurl the truth against falsehood and it cuts right through it and it vanishes clean away! Woe without end for you for what you portray! (Surat al-Anbiya, 18)