"Mother Earth" by Isaac Asimov, first published in Astounding Science Fiction, May 1949, available at the Internet Archive. Here is a plot summary from Wikipedia (emphasis added):
Earth faces a confrontation with its colonies, the "Outer Worlds." A historian looks back and sees the problem beginning a century-and-a-half earlier, when Aurora got permission to "introduce positronic robots into their community life." No date is given, but fifty years before the story starts, the Outer Worlds established an immigration quota against incoming Terran citizens. The balance of power then tipped. Now war appears likely, and there are rumors that Earth has developed an unknown weapon, code-named the "Pacific Project."
On Aurora, there is also concern, but the Aurorans decide that the threat cannot be serious. They use authoritarian methods to suppress Ion Mereanu and his Conservatives, who wish to help Earth. They then call a gathering on Hesperus, one of the Outer Worlds, to unite them against Earth.
There is some rivalry from two other planets, Rhea and Tethys. "All three planets were identically racist, identically exclusivist. Their views on Earth were similar, and completely compatible... But Aurora was the oldest of the Outer Worlds, the most advanced, the strongest militarily... Rhea and Tethys served as a focal point for those who did not recognize Auroran leadership." But Earth unexpectedly sends a threatening message to all of the worlds, uniting them against Earth.
War follows (later termed the "Three-Week War" by historians), and Earth swiftly loses. Trade is ended—the Outer Worlds have no need of Earth's exports, which are mostly agricultural. Earthmen are not allowed to journey beyond the Solar System.
The war was planned in the expectation of defeat—that was what the "Pacific Project" was all about. This is in part to force Earth to make necessary reforms, the use of robots, hydroponic agriculture, and population control. But the Outer Worlds will also weaken and split, because their worlds are biologically ill-suited to long-term human cultures. Several consequences for Earth are predicted from the entire conflict:
We will have a century of rebuilding and revitalization, and at the end of it, we shall face an outer Galaxy which will either be dying or changed. In the first case, we will build a second Terrestrian Empire, more wisely and with greater knowledge than we did the first; one based on a strong and modernized Earth.
In the second case, we will face perhaps ten, twenty, or even all fifty Outer Worlds, each with a slightly different variety of Man. Fifty humanoid species, no longer united against us, each increasingly adapted to its own planet, each with a sufficient tendency towards atavism to love Earth, to regard it as the great and original Mother.
And racism will be dead, for variety will then be the great fact of Humanity, and not uniformity. [...] Mother Earth will finally have given birth not to merely a Terrestrian, but to a Galactic Empire.