Only if you are very selective and pick points that suit your theory and ignore others.

You went from Babylon to Egypt to Greece. What about the Indus Valley Civilization? It was bigger than Egypt and Babylon combined. More people lived and died there than in Egypt and Babylon.

From Rome you skip ahead directly to France/England. What about the Arabs? What about Mongols? What about India/China? What about Turks?

You need to define "dominant culture" a lot better, because it seems that any definition that includes the places you mentioned should also include a lot more that you missed.

The idea of a "dominant culture of the world" cannot defined by any academic means. This is particularly true for the pre-modern era, when the influence of a single culture never approached global scale. Does it mean most populous polity? Most spoken language? Most territory that considered itself a part of said culture?

But even if we were to accept that those cultures were in fact dominant, the peaks of Rome and of France/England are centuries apart, leaving room for powers in Mesoamerica, the Andes, China, and the Middle East (at least) to claim top spot, however we choose to define that.

Yeah, if you selectively cherry pick examples you get trends like this.

Another example of this kind of theory would be of Arnold Toynbee's thesis is that power dissipates towards the fringes of civilization.

For instance civilization first rose in the Levant, before the center being transferred to Greece. Rome was on the edge of the Hellenistic world and it ended up dominating. France/England/Western Europe was on the fringe of the Roman world it they ended up dominating. Finally, the US and Russia ended up being dominating when they started off as the fringes of western civilization.

Keep in mind though that Toynbee is completely discredited, so much so I actually hesitated to even make this post. But it's something fun to think about.