This idea comes from words attributed to Muhammad in a hadith.
“We were with Allah’s Messenger (ﷺ) in an expedition that there came a people to Allah’s Apostle (ﷺ) from the direction of the west. They were dressed in woollen clothes and they stood near a hillock and they met him as Allah’s Messenger (ﷺ) was sitting. I said to myself: Better go to them and stand between him and them that they may not attack him. Then I thought that perhaps there had been going on secret negotiation amongst them. I however, went to them and stood between them and him and I remember four of the words (on that occasion) which I repeat (on the fingers of my hand) that he (Allah’s Messenger) said: You will attack Arabia and Allah will enable you to conquer it, then you would attack Persia and He would make you to conquer it. Then you would attack Rome and Allah will enable you to conquer it, then you would attack the Dajjal and Allah will enable you to conquer him. Nafi’ said: Jabir, we thought that the Dajjal would appear after Rome (Syrian territory) would be conquered.” (Sahih Muslim 6930)
“That means the entire world, and we will conquer it all and rule it through the justice of Islam!”
The “justice of Islam” means Sharia, which in all its various forms denies basic rights to women, non-Muslims, and others.
“Al-Aqsa Mosque Address by Sheikh Muhammad Ayed: It Is Time to Announce Caliphate, Liberate Jerusalem, Set out for Rome and the White House, Conquer the World,” MEMRI, February 4, 2020:
Palestinian cleric Muhammad Ayed “Abu Abdallah” said in an address at the Al-Aqsa Mosque that was uploaded to the Internet on February 4, 2020 that it is time for the “blood and pride of Islam” to stir in the loyal officers and soldiers (in the armies of Muslim countries) and for them to retaliate against “those evil peddlers who make deals,” to pulverize them with a resounding slap, to announce to establishment of a Caliphate, to liberate Jerusalem and the Al-Aqsa Mosque, and to set out for Rome and the White House in order to conquer the entire world and rule it through the justice of Islam, as the Prophet Muhammad had promised.
Muhammad Ayed: “Oh Muslims, there is good in you. It is high time that you revealed it. Oh loyal officers and soldiers, it is high time that the blood and pride of Islam stirred in you, that you set out for Jerusalem and retaliate against those evil peddlers, who make deals. Retaliate for their deal with a resounding slap that will pulverize them. Announce [the establishment] of the Caliphate! March towards Jerusalem! March towards the Al-Aqsa Mosque and liberate it! Retaliate against them and set out for Rome, and then set out for the White House, to that Trump, to Putin, and to others! Let’s conquer the Earth, as the Prophet Muhammad promised: ‘Allah drew the ends of the world near one another for my sake, and I have seen its eastern and western ends. The dominion of my nation would reach those ends that have been drawn near me.’ That means the entire world, and we will conquer it all and rule it through the justice of Islam!”