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: